Construction of Supplemental Functions for Direct Serendipity and Mixed Finite Elements on Polygons

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Construction of Supplemental Functions for Direct Serendipity and Mixed Finite Elements on Polygons

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Submitted:

17 October 2023

Posted:

18 October 2023

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Abstract

New families of direct serendipity and direct mixed finite elements on general planar, strictly convex polygons were recently defined by the authors. The finite elements of index r are H1 and H(div) conforming, respectively, and approximate optimally to order r+1 while using the minimal number of degrees of freedom. The shape function space consists of the full set of polynomials defined directly on the element and augmented with a space of supplemental functions. The supplemental functions were constructed as rational functions, which can be difficult to integrate accurately using numerical quadrature rules when the index is high. This can result in a loss of accuracy in certain cases. In this work, we propose alternative ways to construct the supplemental functions on the element as continuous piecewise polynomials. One approach results in supplemental functions that are in Hp for any p≥1. We prove the optimal approximation property for these new finite elements. We also perform numerical tests on them, comparing results for the original supplemental functions and the various alternatives. The new piecewise polynomial supplements can be integrated accurately, and therefore show better robustness with respect to the underlying meshes used.

Keywords:

Subject: Computer Science and Mathematics - Computational Mathematics

1. Introduction

There has been strong interest in using polygonal and polyhedral meshes when solving certain types of problems via the finite element method. For just a few examples, we note problems in solid mechanics [1,2], elasticity [3,4], fracture mechanics [5,6,7], thin plates [8], porous media [9], topology optimization [10,11,12], and finding eigenvalues [13]. In fact, polygonal meshes are an important motivation for the development and use of methods beyond the classic finite element method, which include, for example, the discontinuous Galerkin methods (including weak Galerkin [14] and ultra-weak methods [15,16,17]), mimetic methods [18,19,20], and virtual element methods [21,22,23,24].

Classic conforming finite element methods have also been developed for use on polygonal meshes, and especially for quadrilateral meshes. Approaches taken include the use of maps from reference finite elements [25,26,27], restriction to low order elements [28,29,30,31], the use of macro-elements [32], basis function enrichment [33,34,35], and construction using barycentric coordinates [36,37,38]. Ideally, we would have families of conforming finite elements defined for any order of accuracy. These would possess a minimal number of degrees of freedom (DoFs) subject to conformity and accuracy constraints. Finite elements based on the use of non-affine maps from reference finite elements display degraded accuracy. Accuracy is restricted if only low order elements are defined. Macro-elements, basis function enrichment, and the use of barycentric coordinates in higher order cases results in finite elements with an excess number of DoFs.

Families of conforming finite elements defined on polygons that maintain both accuracy and a minimal number of DoFs have appeared recently [39,40,41,42] (as well as some finite elements in three dimensions [43,44,45]). The approach taken is to begin with the space of polynomials

P_{r} (E)

of degree up to r defined directly on the physical element E to achieve accuracy of order

r + 1

. To achieve conformity, one then adds in a space of supplemental functions. A basis for the supplemental functions must have certain properties on

\partial E

, but they must be defined over all of E by filling in the interior. The “supplemental function space” is sometimes called the “filling space.”

In this paper, we discuss the construction of the supplemental functions, in the context of the finite elements developed by the current authors in [42], which are called direct finite elements. Let the element

E = E_{N} \subset R^{2}

be a closed, nondegenerate, convex polygon with

N \geq 3

edges. The direct serendipity finite elements of index

r \geq 1

are

H^{1}

-conforming and take the form

{DS}_{r} (E_{N}) = P_{r} (E_{N}) \oplus S_{r}^{DS} (E_{N}),

(1)

where

S_{r}^{DS} (E_{N})

is the space of supplemental functions. The direct mixed finite elements are

H (div)

-conforming and take two forms,

\begin{matrix} V_{r}^{r} (E_{N}) = P_{r}^{2} (E_{N}) \oplus x {\tilde{P}}_{r} (E_{N}) \oplus S_{r}^{V} (E_{N}), \\ V_{r}^{r - 1} (E_{N}) = P_{r}^{2} (E_{N}) \oplus S_{r}^{V} (E_{N}), \end{matrix}

(2)

for full (

r \geq 0

) and reduced (

r \geq 1

)

H (div)

-approximation, respectively, where

{\tilde{P}}_{r} (E_{N})

are the homogeneous polynomials of (exact) degree r. These two finite elements are related to each other by the finite element exterior calculus [46] through the de Rham complex

R ⟶ H^{1} \overset{curl}{\to} H (div) \overset{div}{\to} L^{2} ⟶ 0,

(3)

resulting in, for

s = r - 1, r (s \geq 0)

R ⟶ {DS}_{r + 1} (E_{N}) \overset{curl}{\to} V_{r}^{s} (E_{N}) \overset{div}{\to} P_{s} (E_{N}) ⟶ 0 .

(4)

The consequence is that

\begin{matrix} V_{r}^{r} (E_{N}) = curl {DS}_{r + 1} (E_{N}) \oplus x P_{r}, \\ V_{r}^{r - 1} (E_{N}) = curl {DS}_{r + 1} (E_{N}) \oplus x P_{r - 1}, \end{matrix}

(5)

and, therefore,

S_{r}^{V} (E_{N}) = curl S_{r + 1}^{DS} (E_{N}) .

(6)

The original construction of supplemental functions made use of rational functions (see (21)), which are difficult to numerically integrate accurately. In this work, we introduce two constructions of the supplemental functions

S_{r}^{DS} (E_{N})

which involve using continuous piecewise polynomials. Such constructions are motivated by the work of Kuznetsov and Repin [32], and suggested by the work of Cockburn and Fu [40]. These new supplemental functions can then be accurately integrated by quadrature rules. (A similar, but more complex, construction in three dimensions for cuboidal hexahedra is discussed in [45].)

In the next two sections we present some basic notation and review the general definition of the original direct serendipity and direct mixed finite elements, which have supplemental functions that are

C^{\infty}

-smooth. Our new families of direct finite elements based on piecewise continuous supplemental functions are given in Section 4. We give two constructions of the supplemental functions, so that one set lies in

H^{1}

and the other in

H^{p}

for any integer

p \geq 1

. The approximation properties of these new direct finite elements are given in Section 5. The results are optimal, up to the bounding constant. The proof follows that given in [42], and we concentrate on the modifications that are required to handle the new supplements. In Section 6, we present numerical tests that compare the errors and convergence rates of the new and original direct finite elements. We conclude the paper in Section 7.

2. Notation

We choose to identify the edges and vertices of

E_{N}

adjacently in the counterclockwise direction, as depicted in Figure 1 (throughout the paper, we interpret indices modulo N). Let the edges of

E_{N}

be denoted

e_{i}

i = 1, 2, \dots, N

, and the vertices be

x_{v, i} = e_{i} \cap e_{i + 1}

. Let

ν_{i}

denote the unit outer normal to edge

e_{i}

, and let

τ_{i}

denote the unit tangent vector of

e_{i}

oriented in the counterclockwise direction, for

i = 1, 2, \dots, N

For any two distinct points

y_{1}

and

y_{2}

, let

L [y_{1}, y_{2}]

be the line passing through

y_{1}

and

y_{2}

, and take

ν [y_{1}, y_{2}]

to be the unit vector normal to this line interpreted as going from

y_{1}

y_{2}

and then spinning 90 degrees in the clockwise direction (i.e., pointing to the right). Then we define a linear polynomial giving the signed distance of

x

L [y_{1}, y_{2}]

λ [y_{1}, y_{2}] (x) = - (x - y_{2}) \cdot ν [y_{1}, y_{2}] .

(7)

To simplify the notation for linear functions that will be used throughout the paper, let

L_{i} = L [x_{v, i - 1}, x_{v, i}]

be the line containing edge

e_{i}

and let

λ_{i} (x)

give the distance of

x \in R^{2}

to edge

e_{i}

opposite the normal direction, i.e.,

\begin{matrix} λ_{i} (x) & = λ [x_{v, i - 1}, x_{v, i}] (x) = - (x - x_{v, i}) \cdot ν_{i}, i = 1, 2, \dots, N . \end{matrix}

(8)

These functions are strictly positive in the interior of

E_{N}

, and

λ_{i}

vanishes on the edge

e_{i}

3. Direct serendipity and mixed finite elements

The general development of direct serendipity and mixed finite elements is given in [42]. The definition of the supplemental space

S_{r}^{DS} (E_{N})

in (1) is key to the construction. For completeness, we review the definitions of these direct finite elements here.

When

N = 3

(triangles), the direct serendipity supplemental space

S_{r}^{DS} (E_{3})

is empty. When

N \geq 4

and

1 \leq r < N - 2

, the direct serendipity spaces

{DS}_{r} (E_{N})

are defined as subspaces of

{DS}_{N - 2} (E_{N})

by the rule

\begin{matrix} {DS}_{r} (E_{N}) = \{φ \in {DS}_{N - 2} (E_{N}) {: φ |}_{e} \in P_{r} (e) for all edges e of E_{N}\} . \end{matrix}

(9)

Therefore, we only need to understand

S_{r}^{DS} (E_{N})

for

r \geq N - 2

and

N \geq 4

To define the supplemental basis functions, two series of choices must be made for each

i, j

such that

1 \leq i < j \leq N

and

2 \leq j - i \leq N - 2

(i.e.,

e_{i}

and

e_{j}

are nonadjacent edges). First, as shown in Figure 2, one must choose two distinct points

x_{i, j}^{1} \in L_{i}

and

x_{i, j}^{2} \in L_{j}

that avoid the intersection point

x_{i, j} = L_{i} \cap L_{j}

, if it exists. Then let

λ_{i, j} (x) = λ [x_{i, j}^{1}, x_{i, j}^{2}] (x) = - (x - x_{i, j}^{2}) \cdot ν_{i, j}, ν_{i, j} = ν [x_{i, j}^{1}, x_{i, j}^{2}],

(10)

be the linear function associated to the line

L_{i, j} = L [x_{i, j}^{1}, x_{i, j}^{2}]

. Second,

R_{i, j}

must be chosen to satisfy

\begin{matrix} R_{i, j} {(x) |}_{e_{i}} & = - 1, & R_{i, j} {(x) |}_{e_{j}} & = 1 . \end{matrix}

(11)

The supplemental space for

r \geq N - 2 \geq 2

is of the form

\begin{matrix} S_{r}^{DS} (E_{N}) & = span \{ϕ_{s, i, j} : 1 \leq i < j \leq N, 2 \leq j - i \leq N - 2\}, \end{matrix}

(12)

\begin{matrix} ϕ_{s, i, j} & = (\prod_{k \neq i, j} λ_{k}) λ_{i, j}^{r - N + 2} R_{i, j} . \end{matrix}

(13)

3.1. Direct serendipity finite elements

Every shape function of the direct serendipity finite element

{DS}_{r} (E_{N})

is a sum of a polynomial and a linear combination of the supplemental functions, as in (1). To implement them, one must define the DoFs. For example, for

ψ \in {DS}_{r} (E_{N})

, one can take

\begin{matrix} ψ (x_{v, i}), & \forall i = 1, 2, \dots, N, \end{matrix}

(14)

\begin{matrix} \int_{e_{i}} ψ p d σ, & \forall p \in P_{r - 2} (e_{i}), i = 1, 2, \dots, N, \end{matrix}

(15)

\begin{matrix} \int_{E_{N}} ψ q d x, & \forall q \in P_{r - N} (E_{N}), \end{matrix}

(16)

where

d σ

is the one dimensional surface measure. Alternatively, one can use nodal DoFs (i.e., evaluation at a node point) in place of () and/or (). For the former, on each edge

e_{i}

, its corresponding edge nodes are

r - 1

points such that they, along with the two vertices, are equally distributed on

e_{i}

. For the latter, the interior cell nodes can be set to be the Lagrange nodes of order

r - N

of a triangle that lies strictly inside

E_{N}

The basis of

{DS}_{r} (E_{N})

corresponding to the DoFs can be constructed. Given a computational mesh of convex polygons

T_{h}

over a domain

Ω

, the basis can be simply pieced together to form a global

H^{1}

-conforming basis of the space

{DS}_{r} (Ω) \subset H^{1} (Ω)

3.2. Direct mixed finite elements

As discussed in the introduction, full,

V_{r}^{r} (E_{N})

, and reduced,

V_{r}^{r - 1} (E_{N})

H (div)

-approximating mixed finite element spaces follow from a de Rham complex (3), where the direct serendipity finite elements serve as the precursor (4). The supplemental space is related to

S_{r + 1}^{DS} (E_{N})

by the simple formula (6).

The DoFs for these spaces (with

s = r \geq 0

s = r - 1 \geq 0

) can be taken to be

\begin{matrix} \int_{e_{i}} v \cdot ν_{i} p d σ, & \forall p \in P_{r} (e_{i}), i = 1, 2, \dots, N, \end{matrix}

(17)

\begin{matrix} \int_{E_{N}} v \cdot \nabla q d x, & \forall q \in P_{s} (E_{N}), q not constant, \end{matrix}

(18)

\begin{matrix} \int_{E_{N}} v \cdot ψ d x, & \forall ψ \in B_{r}^{V} (E_{N}), if r \geq N - 1, \end{matrix}

(19)

where the

H^{1} (E_{N})

and

H (div; E_{N})

bubble functions, for

r \geq N - 1

, are

B_{r + 1} (E_{N}) = λ_{1} λ_{2} \dots λ_{N} P_{r - N + 1} (E_{N}) and B_{r}^{V} (E_{N}) = curl B_{r + 1} (E_{N}) .

(20)

Given the mesh

T_{h}

over

Ω

, one constructs the basis and the

H (div)

-conforming global space

V_{r}^{s} (Ω) \subset H (div)

(see [42] for details). As an alternative, when solving partial differential equations, one can use the hybrid form of the method [47], which does not require the construction of global basis functions.

4. Piecewise continuous supplements

In [42],

R_{i, j}

satisfying (11) on

E_{N}

, for

1 \leq i < j \leq N

2 \leq j - i \leq N - 2

, was taken to be the simple rational function

R_{i, j}^{rational} (x) = \frac{λ_{i} (x) - λ_{j} (x)}{λ_{i} (x) + λ_{j} (x)} .

(21)

These rational functions are smooth over the element. We now give new direct serendipity and mixed finite elements by providing an alternate construction of

R_{i, j}

as a piecewise continuous polynomial defined over a sub-partition of

E_{N}

. We present two strategies, the first of which is convenient for the construction of continuous supplemental functions in

H^{1} (E_{N})

, and the second for constructing smoother supplemental functions in

H^{p} (E_{N})

for integer

p \geq 1

4.1. Supplemental functions in $H^{1} (E_{N})$

Our first strategy for constructing

R_{i, j}

requires a sub-triangulation of the element

E_{N}

, and we present two natural choices. The first sub-triangulation is depicted in Figure 3 and denoted as

T^{n} (E_{N})

. One picks a vertex

x_{v, n}

and divides

E_{N}

into

N - 2

sub-triangles. The sub-triangles are

T_{m}^{n}

with vertices

x_{v, n}

x_{v, m}

, and

x_{v, m + 1}

, where

m = n + 1, \dots, n + N - 2

. For the second sub-triangulation, depicted in Figure 4 and denoted as

T^{x_{c}} (E_{N})

, one picks a point

x_{c}

in the interior of

E_{N}

and divides it into N sub-triangles. Now the sub-triangles are

T_{m}^{x_{c}}

with vertices

x_{c}

x_{v, m}

, and

x_{v, m + 1}

, where

m = 1, 2, \dots, N

. We use the centroid of the element for

x_{c}

Let the piecewise polynomial function space of degree s corresponding to each sub-triangulation be

\begin{matrix} P_{s} (T^{n} (E_{N})) & = {f \in C^{0} (E) | f |_{T_{m}^{n}} \in P_{s} (T_{m}^{n}), m = n + 1, \dots, n + N - 2}, \end{matrix}

(22)

\begin{matrix} P_{s} (T^{x_{c}} (E_{N})) & = {f \in C^{0} (E) | f |_{T_{m}^{x_{c}}} \in P_{s} (T_{m}^{x_{c}}), m = 1, 2, \dots, N} . \end{matrix}

(23)

We construct

R_{i, j}

P_{1} (T^{n} (E_{N}))

P_{1} (T^{x_{c}} (E_{N}))

, depending on which of the two sub-triangulations is used, such that

\begin{matrix} R_{i, j} |_{e_{i}} = - 1, R_{i, j} {|_{e_{j}} = 1, R_{i, j} |}_{v_{k}} = 0, \forall k \neq i - 1, i, j - 1, j . \end{matrix}

(24)

by using interpolation at the vertices of the sub-triangles. If the sub-triangulation is chosen to be

T^{n} (E_{N})

, the restrictions (24) uniquely specify all the vertex values. However, if the triangulation is

T^{x_{c}} (E_{N})

, the center value is not determined, so we assign

R_{i, j} (x_{c}) = 0

Our construction has

R_{i, j}

being

- 1

e_{i}

and 1 on

e_{j}

as required by (11). Moreover,

R_{i, j} \in H^{1} (E_{N})

. After constructing the supplemental functions in () with this

R_{i, j}

, each

ϕ_{s, i, j}

is in

P_{r + 1} (T^{n} (E_{N}))

P_{r + 1} (T^{x_{c}} (E_{N}))

, and therefore also in

H^{1} (E_{N})

4.2. Supplemental functions in $H^{p} (E_{N})$

We now present the second of our two strategies for constructing

R_{i, j}

for two nonadjacent edges

e_{i}

and

e_{j}

. Recall that

λ_{k} (x)

is the linear polynomial giving the (signed) distance to the line

L_{k}

extending edge

e_{k}

. When

e_{i}

and

e_{j}

are parallel, we simply define

R_{i, j}

as the linear polynomial

R_{i, j} = \frac{λ_{i} - λ_{j}}{λ_{i} (x_{v, j})} .

(25)

When

e_{i}

and

e_{j}

are not parallel, we first define a sub-partition of

E_{N}

by adding a single extra line

ℓ^{i, j}

through a point

x^{i, j}

as depicted in Figure 5. The point

x^{i, j}

is chosen so that it is closer to

L_{j}

than the endpoints of

e_{i}

, i.e.,

\begin{matrix} λ_{j} (x^{i, j}) \leq min {λ_{j} (x_{v, i - 1}), λ_{j} (x_{v, i})} . \end{matrix}

(26)

The line

ℓ^{i, j}

passes through

x^{i, j}

and is parallel to

e_{j}

. This line divides

E_{N}

into

E_{N}^{i, j, 1}

near

e_{i}

and

E_{N}^{i, j, 0}

near

e_{j}

, i.e.,

\begin{matrix} E_{N}^{i, j, 1} & = E_{N} \cap {x | λ_{j} (x) \geq λ_{j} (x^{i, j})}, \end{matrix}

(27)

\begin{matrix} E_{N}^{i, j, 0} & = E_{N} \cap {x | λ_{j} (x) < λ_{j} (x^{i, j})} . \end{matrix}

(28)

Let

ν^{i, j} = - ν_{j}

be the unit normal vector of

ℓ^{i, j}

pointing into

E_{N}^{i, j, 1}

, and let

τ^{i, j} = τ_{j}

be a unit tangent vector.

We next construct the function

ρ^{i, j}

, which is 1 on edge

e_{i}

and 0 on edge

e_{j}

. It is defined piecewise on the sub-partition of

E_{N}

\begin{matrix} ρ^{i, j} (x) = \{\begin{matrix} 1, & x \in E_{N}^{i, j, 1}, \\ 1 - {(1 - \frac{λ_{j} (x)}{λ_{j} (x^{i, j})})}^{p}, & x \in E_{N}^{i, j, 0}, \end{matrix} \end{matrix}

(29)

where

p \geq 1

is an integer. The function is continuous, since

λ_{j} (x) = λ_{j} (x^{i, j})

ℓ^{i, j}

implies that

ρ^{i, j} |_{ℓ^{i, j}} = 1

in either case of the definition. Moreover, in the tangential direction,

\begin{matrix} \frac{\partial ρ^{i, j}}{\partial τ^{i, j}} |_{ℓ^{i, j}} = 0, \end{matrix}

(30)

and, in the normal direction,

\begin{matrix} \frac{\partial ρ^{i, j}}{\partial ν^{i, j}} (x) = \{\begin{matrix} 0, & x \in E_{N}^{i, j, 1}, \\ \frac{p}{λ_{j} (x^{i, j})} {(1 - \frac{λ_{j} (x)}{λ_{j} (x^{i, j})})}^{p - 1}, & x \in E_{N}^{i, j, 0}, \end{matrix} \end{matrix}

(31)

which is continuous for

p > 1

, so

ρ^{i, j} \in C^{1} (E_{N})

. By iterating the argument, we have that

ρ^{i, j} \in C^{p - 1} (E_{N})

and so also in

H^{p} (E_{N})

for

p > 1

. If

p = 1

ρ^{i, j}

is continuous, so it is in

H^{1} (E_{N})

Finally, after constructing both

ρ^{i, j}

and

ρ^{j, i}

, we define

R_{i, j} = ρ^{j, i} - ρ^{i, j},

(32)

which is

- 1

e_{i}

, 1 on

e_{j}

. Moreover,

R_{i, j} \in H^{p} (E_{N})

. The supplemental functions in () constructed with this

R_{i, j}

lie in

H^{p} (E_{N})

We end this section with two specific examples, using the sub-partitions shown in Figure 6, which divide

E_{N}

by N lines. The first example has a sub-partition based on the midpoints

x_{e, i}^{M}

of the edges

e_{i}

i = 1, 2, \dots, N

) and gives rise to the spaces denoted

{DS}_{r}^{M} (E_{N})

and

V_{r}^{M, s} (E_{N})

. We compute the minimal distance of the midpoints to the edges, i.e.,

h_{M} = min_{1 \leq i \leq N, k = i \pm 1} λ_{i} (x_{e, k}^{M}) .

(33)

Then for any two non parallel and nonadjacent edges

e_{i}

and

e_{j}

, simply take the partition line

ℓ^{i, j}

to be the line parallel to

e_{j}

that is the fixed distance

h_{M} > 0

away and intersects

E_{N}

The second specific example uses a sub-partition based on trisecting each edge, resulting in the points, for edge

e_{i}

, being denoted counterclockwise as

x_{e, i, k}

for

k = 1, 2

i = 1, 2, \dots, N

. In this case, we simply take

x^{i, j}

to be the closest of these points to

L_{j}

, omitting

x_{e, j, 1}

and

x_{e, j, 2}

. We denote the resulting spaces

{DS}_{r}^{T} (E_{N})

and

V_{r}^{T, s} (E_{N})

5. Approximation properties

We discuss now the global approximation properties for our direct finite element spaces. The results of [42] do not directly apply here because there it was assumed that the functions

R_{i, j}

are smooth on the element. Consider a collection of meshes

T_{h}

of convex polygons partitioning a domain

Ω

, where

h > 0

is the maximal element diameter.

We need to make the usual assumption that our collection of meshes is uniformly shape regular ([48], pp. 104–105). For any

E_{N} \in T_{h}

, let

h_{E_{N}}

be its diameter. Denote by

T_{i}

i = 1, 2, \dots, N (N - 1) (N - 2) / 6

, the sub-triangle of

E_{N}

with vertices being three of the N vertices of

E_{N}

, and define

ρ_{E_{N}} = 2 min_{1 \leq i \leq N (N - 1) (N - 2) / 6} {diameter of the largest circle inscribed in T_{i}} .

(34)

The shape regularity parameter of the single mesh

T_{h}

σ_{T_{h}} = min_{E_{N} \in T_{h}} \frac{ρ_{E_{N}}}{h_{E_{N}}} .

(35)

Assumption 1.

The collection of meshes

{T_{h}}_{h > 0}

isuniformly shape regular.That is, the shape regularity parameters are bounded below by a positive constant: there exists

σ_{*} > 0

, independent of

T_{h}

and

h > 0

, such that the ratio

\frac{ρ_{E_{N}}}{h_{E_{N}}} \geq σ_{*} > 0 for all E_{N} \in T_{h}, h > 0 .

(36)

We also require some mild restrictions on the construction of

S_{r}^{DS} (E_{N})

Assumption 2.

For every

E_{N} \in T_{h}

, assume that the functions of

S_{r}^{DS} (E_{N})

are constructed using

λ_{i, j}

such that the zero set

L_{i, j}

intersects

e_{i}

and

e_{j}

. Moreover, suppose that

R_{i, j} \in H^{p} (E_{N})

for some

p \geq 1

and that the sub-partitions introduced in Section 4 for their construction depend continuously on the vertices of

E_{N}

The continuous dependence requirement of the sub-partitions is met if we systematically choose the points

x_{c}

in Section 4.1 (say as the centroid) and

x^{i, j}

satisfying (26) in Section 4.2 (say be taking

x^{i, j}

as the closer endpoint of

e_{i}

L_{j}

, or so that

λ_{j} (x^{i, j}) = \frac{1}{2} min {λ_{j} (x_{v, i - 1}), λ_{j} (x_{v, i})}

We state first the approximation result for

{DS}_{r} (Ω)

Theorem 1.

Let

r \geq 1

1 \leq p \leq \infty

, and

ℓ > 1 / p

(or

ℓ \geq 1

p = 1

). If Assumptions 1–2 hold (so the basis functions are in

H^{p}

on each element), then there exists a constant

C > 0

, such that for all functions

v \in W^{ℓ, p} (Ω)

\begin{matrix} inf_{v_{h} \in {DS}_{r} (Ω)} ∥ v - v_{h} ∥_{W^{m, p} (Ω)} \leq C h^{ℓ - m} {∥ v ∥}_{W^{ℓ, p} (Ω)}, 0 \leq ℓ \leq r + 1, m = 0, 1 . \end{matrix}

(37)

Proof.

The methodology of the proof follows [41,42]. The key difference is that we must relax the smoothness requirement made on the supplemental functions. We highlight the differences, and leave the reader to consult [41,42] for some of the details.

Given a mesh

T_{h}

, we construct an interpolation operator

I_{h}^{r} : W_{p}^{l} (Ω) \to {DS}_{r}

as a generalization of that defined in [49]. To do so, we use a nodal set of DoFs for the finite elements, and identify global nodal points

a_{i}

i = 1, 2, \dots, dim {DS}_{r}

. These nodal points must be chosen systematically with respect to the vertices of the mesh, so they depend continuously on them. The global nodal basis function for

a_{i}

is denoted

φ_{i}

A geometry object

K_{i}

is associated to each

a_{i}

. If

a_{i}

lies in the interior of some element, we choose the element to be

K_{i}

. Otherwise, we choose an edge containing

a_{i}

to be

K_{i}

, where we additionally ask that

K_{i} \subset \partial Ω

a_{i} \in \partial Ω

. We use these to define the dual basis

ψ_{i}

with respect to

L^{2} (K_{i})

i = 1, 2, \dots, dim {DS}_{r}

. The corresponding interpolation operator

I_{h}^{r} : W_{p}^{l} (Ω) \to {DS}_{r}

is then

I_{h}^{r} v (x) = \sum_{i = 1}^{dim {DS}_{r}} (\int_{K_{i}} ψ_{i} (y) v (y) d y) φ_{i} (x) .

(38)

There are two essential steps towards showing the approximation property. First, the nodal basis functions are bounded,

max_{1 \leq i \leq dim {DS}_{r} (Ω)} max_{E \in T_{h}} | | φ_{i} {| |}_{W_{q}^{m} (E)} \leq C,

(39)

and, second, the dual basis functions are bounded up to a scaling factor,

| | ψ_{i} {| |}_{L^{\infty} (K_{i})} \leq C h_{K_{i}}^{- dim K_{i}} .

(40)

We show the necessary boundedness by mapping the elements and using a continuity and compactness argument.

As depicted in Figure 7, to each element

E_{N}

, we associate a regular polygon (equilateral and equiangular)

{\hat{E}}_{N}

. We can then define a map

F_{E_{N}} : {\hat{E}}_{N} \to E_{N}

as a composition of a map that changes the geometry but not the size to

{\tilde{E}}_{N}

, and then a scaling map (see [42] for precise details).

Define the nodal basis functions

φ_{i}^{{\tilde{E}}_{N}}

{\tilde{E}}_{N}

. It is enough to show the boundedness of their

W_{q}^{m}

norms. Although they are no longer smooth functions, compared to [41,42], they are continuous on

{\tilde{E}}_{N}

, and smooth on all the subregions generated by the sub-partition. Moreover, by assumption the sub-partition is required to depend continuously on the vertices of

{\tilde{E}}_{N}

. Therefore,

φ_{i}^{{\tilde{E}}_{N}}

will still depend continuously on

\hat{x} = F_{{\tilde{E}}_{N}}^{- 1} (\tilde{x})

and the vertices of

{\tilde{E}}_{N}

, which vary in a compact set. We conclude that the nodal basis functions are bounded in

W_{q}^{m}

norm. The boundedness of

ψ_{i}

in the

L^{\infty}

norm can be shown in a similar way. □

For the mixed finite elements, we have the following result, wherein we see projection operators

π : H (div; Ω) \cap {(L^{2 + ϵ} (Ω))}^{2} \to V_{r}^{s}

s = r - 1, r

, where

ϵ > 0

, and

P_{W_{s}}

, the

L^{2}

-orthogonal projection operator onto

W_{s} = \nabla \cdot V_{r}^{s}

Theorem 2.

Let

r = s \geq 0

r \geq 1

s = r - 1

. If Assumptions A1–A2 hold, then there is a constant

C > 0

, such that

\begin{matrix} {∥ v - π v ∥}_{L^{2} (Ω)} & \leq {C ∥ v ∥}_{H^{k} (Ω)} h^{k}, & k = 1, \dots, r + 1, \end{matrix}

(41)

\begin{matrix} ∥ p - P_{W_{s}} {p ∥}_{L^{2} (Ω)} & \leq {C ∥ p ∥}_{H^{k} (Ω)} h^{k}, & k = 0, 1, \dots, s + 1, \end{matrix}

(42)

\begin{matrix} {∥ \nabla \cdot (v - π v) ∥}_{L^{2} (Ω)} & \leq {C ∥ \nabla \cdot v ∥}_{H^{k} (Ω)} h^{k}, & k = 0, 1, \dots, s + 1, \end{matrix}

(43)

where

s = r - 1 \geq 0

and

s = r \geq 1

for reduced and full

H (div)

-approximation, respectively. Moreover, the discrete inf-sup condition

sup_{v_{h} \in V_{r}^{s}} \frac{(w_{h}, \nabla \cdot v_{h})}{∥ v_{h} ∥_{H (div)}} \geq γ {∥ w_{h} ∥}_{L^{2} (Ω)}, \forall w_{h} \in W_{s},

(44)

holds for some

γ > 0

independent of

h > 0

For the proof, we define the projection operator

π

by piecing together local operators

π_{E}

that are defined in terms of the DoFs (17)–(). The approximation properties given in [41,42] hold with a similar proof, using now that the subregions generated by the sub-partition depend continuously on the vertices of the element.

6. Numerical results

We present numerical experiments for our new finite elements as applied to Poisson’s equation

\begin{matrix} - \nabla \cdot (\nabla p) & = f & in Ω, \end{matrix}

(45)

\begin{matrix} p & = 0 & on \partial Ω, \end{matrix}

(46)

where

f \in L^{2} (Ω)

. The corresponding weak form finds

p \in H_{0}^{1} (Ω)

such that

\begin{matrix} (\nabla p, \nabla q) = (f, q), \forall q \in H_{0}^{1} (Ω), \end{matrix}

(47)

where

(\cdot, \cdot)

is the

L^{2} (Ω)

inner product. Setting

u = - \nabla p,

(48)

we have the mixed weak form, which finds

u \in H (div; Ω)

and

p \in L^{2} (Ω)

such that

\begin{matrix} (u, v) - (p, \nabla \cdot v) & = 0, & \forall v \in H (div; Ω), \end{matrix}

(49)

\begin{matrix} (\nabla \cdot u, w) & = (f, w), & \forall w \in L^{2} (Ω) . \end{matrix}

(50)

These weak forms naturally give rise to finite element approximations. According to Theorems 1 and 2, the following convergence analysis holds by a standard argument [26,50].

Theorem 3.

If Assumptionsn A1–A2 hold, then there exists a constant

C > 0

, independent of

T_{h}

and

h > 0

, such that for

r \geq 1

\begin{matrix} ∥ p - p_{h} ∥_{m, Ω} & \leq C h^{ℓ + 1 - m} {| p |}_{ℓ + 1, Ω}, ℓ = 0, 1, \dots, r, m = 0, 1, \end{matrix}

(51)

where

p_{h} \in {DS}_{r} (Ω) \cap H_{0}^{1} (Ω)

approximates (47). Moreover, with

r = s \geq 0

r \geq 1

s = r - 1

\begin{matrix} ∥ u - u_{h} ∥_{0, Ω} & \leq {C ∥ u ∥}_{k, Ω} h^{k}, & k = 1, \dots, r + 1, \end{matrix}

(52)

\begin{matrix} ∥ p - p_{h} ∥_{0, Ω} & \leq {C ∥ u ∥}_{k, Ω} h^{k}, & k = 1, \dots, s + 1, \end{matrix}

(53)

\begin{matrix} ∥ \nabla \cdot (u - u_{h}) ∥_{0, Ω} & \leq {C ∥ \nabla \cdot u ∥}_{k, Ω} h^{k}, & k = 0, 1, \dots, s + 1, \end{matrix}

(54)

where

(u_{h}, p_{h}) \in V_{r}^{s} \times W_{s}

approximates (49)–().

We perform our tests on a unit square domain

Ω = {[0, 1]}^{2}

, and take the source term

f (x) = 2 π^{2} sin (π x_{1}) sin (π x_{2})

, so the exact solution is

u (x_{1}, x_{2}) = sin (π x_{1}) sin (π x_{2})

. We consider five types of supplemental spaces. The original direct serendipity and mixed finite element spaces will be denoted

{DS}_{r}^{R}

and

V_{r}^{R, s}

, respectively. These use supplements based on the rational functions (21).

For the

H^{1}

supplemental functions introduced in Section 4.1, there are two varieties. Denote the space using supplemental functions that are constructed based on the vertex sub-triangulation as

{DS}_{r}^{V}

and its corresponding mixed spaces as

V_{r}^{V, s}

, and those based on the center point sub-triangulation as

{DS}_{r}^{C}

and

V_{r}^{C, s}

, respectively. The spaces based on the

H^{p}

supplements were described in Section 4.2 and denoted

{DS}_{r}^{M}, V_{r}^{M, s}

and

{DS}_{r}^{T}, V_{r}^{T, s}

6.1. The meshes used

Approximate solutions are computed on a sequence of Voronoi meshes

T_{h}^{2}

generated by the package PolyMesher [51]. Each mesh has

n^{2}

elements, which are generated with

n^{2}

random initial seeds and up to

10^{4}

smoothing iterations to improve the shape regularity.

For comparison to the results appearing in [42], we use the same mesh sequence

T_{h}^{2}

with

n = 6

, 10, 14, 18, and 22. We show the meshes for

n = 6

and

n = 18

in Figure 8. The shape regularity parameters are given in Table 1. Note that the

n = 10

and

n = 18

meshes are the least regular.

In [42] it was observed numerically that the

n = 18

mesh performed well for the original direct finite elements (using rational supplemental functions) when

r = 2, 3, 4

, but had a degraded convergence rate when

r = 5

. The problem was resolved by removing short edges from the

n = 18

mesh. However, as we will see in this section, the problem is actually due to inaccurate numerical quadrature of the rational supplemental functions, which only showed up in those tests for the more refined mesh (i.e., not

n = 10

) and higher values of r.

6.2. Results for direct serendipity spaces

We present and compare the results of the numerical tests performed for

{DS}_{r}^{R}

{DS}_{r}^{V}

{DS}_{r}^{C}

{DS}_{r}^{M}

, and

{DS}_{r}^{T}

, where

r = 2, 3, 4, 5

. We take

p = 1

in Section 4.2 for the construction of

{DS}_{r}^{M}

and

{DS}_{r}^{T}

, because it gives better results than a larger p in most cases. According to Theorem 3, we expect all those spaces to have the convergence rates

r + 1

for

L^{2}

errors and r for

H^{1}

-seminorm errors. As Table 2 and Table 3 suggests, the convergence rates at

n = 10, 14, 22

for

{DS}_{r}^{R}

are all slightly better than optimal in this test. However, we can observe a slower convergence rate at

n = 18

for

{DS}_{r}^{R}

. (We interject that convergence rates are computed as improvement from the previous mesh in our sequence.)

In Table 4, we compare the results for the

n = 18

mesh of

{DS}_{r}^{R}

{DS}_{r}^{V}

{DS}_{r}^{C}

{DS}_{r}^{M}

, and

{DS}_{r}^{T}

. On the one hand, the results suggest that the new spaces are all approximately optimal for

r = 5

, which is an obvious improvement compared to

{DS}_{5}^{R}

. On the other hand, the errors for

r = 2, 3, 4

of the new spaces are slightly worse than those of

{DS}_{r}^{R}

, and among all the new spaces,

{DS}_{r}^{C}

has the best performance in error. We conclude that

{DS}_{r}^{C}

shows the best overall performance among all the spaces considered.

We suggest that the reason of such observation is that the dominant errors for

r = 5

is from the numerical quadrature applied to the integration of rational functions, especially on the elements that are less shape regular. However, for

r = 2, 3, 4

, the new supplements, as piecewise polynomials, cannot approximate the shape of a smooth function as well as the original rational supplements, especially those from

{DS}_{r}^{M}

and

{DS}_{r}^{T}

, of which

R_{i, j}

for

e_{i} ∦ e_{j}

are flat in the middle and oscillate near the boundary. On the contrary, the supplements from

{DS}_{r}^{V}

and

{DS}_{r}^{C}

are more reasonably shaped, and those from

{DS}_{r}^{C}

are better since its partition has sub-triangles that are more shape regular (as was shown in Figure 3 and Figure 4). This argument is also supported by the observation that the results are usually worse if we take larger p for

{DS}_{r}^{M}

and

{DS}_{r}^{T}

, where the shape of the supplements are even worse.

6.3. Results for direct mixed spaces

We perform numerical tests for

V_{r}^{R, s}

V_{r}^{V, s}

V_{r}^{C, s}

V_{r}^{M, s}

V_{r}^{T, s}

, for the full

H (div)

-approximation spaces where

r = s = 0, 1, 2, 3,

and the reduced

H (div)

-approximation spaces where

r = 1, 2, 3

, and

s = r - 1

. Since those mixed spaces are constructed from corresponding direct serendipity spaces

{DS}_{r + 1}

, it is natural that we find the comparison of the results similar to the small r cases discussed in Section 6.2. For all the spaces, we can observe the convergence rates approximately optimal in general but the errors are slightly worse for

n = 18

, especially when

r = s = 3

, as shown in Table 5 and Table 6. All spaces perform similarly well, although

V_{r}^{R, s}

usually performs best in these tests. Among the new spaces,

V_{r}^{C, s}

performs a bit better, and it gives results close to those of

V_{r}^{R, s}

. For reference, we provide the numerical results for

V_{r}^{C, s}

in Table 7 and Table 8.

7. Conclusions

We reviewed the construction of direct serendipity and mixed finite elements on non-degenerate, planar convex polygons. The direct serendipity finite element spaces are of the form

\begin{matrix} {DS}_{r} (E_{N}) = P_{r} (E_{N}) \oplus S_{r}^{DS} (E_{N}) . \end{matrix}

(55)

The full and reduced

H (div)

-approximation mixed finite element spaces are obtained from a de Rham complex, where the direct serendipity finite elements serve as a precursor. The mixed spaces are of the form

\begin{matrix} \begin{matrix} V_{r}^{r} (E_{N}) = P_{r}^{2} (E_{N}) \oplus x {\tilde{P}}_{r} (E_{N}) \oplus S_{r}^{V} (E_{N}), \\ V_{r}^{r - 1} (E_{N}) = P_{r}^{2} (E_{N}) \oplus S_{r}^{V} (E_{N}), \end{matrix} \end{matrix}

(56)

where

S_{r}^{V} (E_{N}) = curl S_{r + 1}^{DS} (E_{N}) .

(57)

We presented two approaches to construct the supplemental functions in

S_{r}^{DS} (E_{N})

as piecewise polynomials. The first approach divides a polygonal element

E_{N}

into sub-triangles, and constructs the supplements as continuous piecewise polynomials that lie in

H^{1} (E_{N})

. The second approach has a more complicated subdivision of

E_{N}

that needs to be treated carefully. However, it provides a framework for constructing supplements that lie in

H^{p} (E_{N})

for any

p \geq 1

The approximation properties of the new finite elements were proved under the regularity assumption of the mesh sequences and some mild restrictions on the construction.

We performed numerical tests on a randomly generated mesh sequence and compared results for five different ways of constructing the supplemental functions, including the original construction using smooth but rational functions. The comparison suggested that it is better to use the piecewise polynomial supplements rather than the rational supplements for higher order r. Although the rational supplements are smooth and so tend to approximate better, noticeable error could be seen due to inaccurate numerical integration, especially on meshes with short edges. Among the new spaces, it was found that the spaces with supplements based on the center point sub-triangulation () performed best.

Author Contributions

Both authors contributed about equally to the work. All authors have read and agreed to the published version of the manuscript.

Funding

This research was funded by U.S. National Science Foundation grant number DMS-2111159.

Data Availability Statement

All pertinent data generated or analyzed during this study are included in this published article. Supporting data omitted from the article are available from the corresponding author on reasonable request.

Conflicts of Interest

The authors declare no conflict of interest.

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Figure 1. A pentagon

E_{5}

, with edges

e_{i}

, outer unit normals

ν_{i}

, tangents

τ_{i}

, and vertices

x_{v, i}

Figure 1. A pentagon

E_{5}

, with edges

e_{i}

, outer unit normals

ν_{i}

, tangents

τ_{i}

, and vertices

x_{v, i}

Figure 2. Illustration on

E_{5}

of the zero line

L_{2, 4}

λ_{2, 4} (x) = - (x - x_{2, 4}^{2}) \cdot ν_{2, 4}

and the intersection point

x_{2, 4} = L_{2} \cap L_{4}

, if it exists.

Figure 2. Illustration on

E_{5}

of the zero line

L_{2, 4}

λ_{2, 4} (x) = - (x - x_{2, 4}^{2}) \cdot ν_{2, 4}

and the intersection point

x_{2, 4} = L_{2} \cap L_{4}

, if it exists.

Figure 3. A sub-triangulation based on a common fixed vertex. Shown is

T^{5} (E_{5})

using the fixed vertex

x_{v, 5}

Figure 3. A sub-triangulation based on a common fixed vertex. Shown is

T^{5} (E_{5})

using the fixed vertex

x_{v, 5}

Figure 4. Sub-triangulation based on a center point. Shown is

T^{x_{c}} (E_{5})

using the centroid

x_{c}

Figure 4. Sub-triangulation based on a center point. Shown is

T^{x_{c}} (E_{5})

using the centroid

x_{c}

Figure 5. A sub-division of

E_{6}

using the line

ℓ^{i, j}

Figure 5. A sub-division of

E_{6}

using the line

ℓ^{i, j}

Figure 6. The two sub-partitions of

E_{5}

used for constructing specific

H^{p}

supplemental functions. The left one is used for

{DS}_{r}^{M}

and

V_{r}^{M, s}

, where

h_{M} = λ_{5} (x_{e, 4}^{M})

. The right one is used for

{DS}_{r}^{T}

and

V_{r}^{T, s}

, where the closest trisection points are marked in red.

Figure 6. The two sub-partitions of

E_{5}

used for constructing specific

H^{p}

supplemental functions. The left one is used for

{DS}_{r}^{M}

and

V_{r}^{M, s}

, where

h_{M} = λ_{5} (x_{e, 4}^{M})

. The right one is used for

{DS}_{r}^{T}

and

V_{r}^{T, s}

, where the closest trisection points are marked in red.

Figure 7. An element

E_{5} \in T_{h}

is shown on the right-hand side in its translated and rotated local coordinates. It is the image of a regular reference polygon

{\hat{E}}_{5}

on the left-hand side. The map is decomposed into one that changes the geometry but not the size

F_{{\tilde{E}}_{5}} : {\hat{E}}_{5} \to {\tilde{E}}_{5}

, and a scaling map

\tilde{x} \mapsto H \tilde{x}

Figure 7. An element

E_{5} \in T_{h}

is shown on the right-hand side in its translated and rotated local coordinates. It is the image of a regular reference polygon

{\hat{E}}_{5}

on the left-hand side. The map is decomposed into one that changes the geometry but not the size

F_{{\tilde{E}}_{5}} : {\hat{E}}_{5} \to {\tilde{E}}_{5}

, and a scaling map

\tilde{x} \mapsto H \tilde{x}

Figure 8. Meshes with

6 \times 6

and

18 \times 18

elements.

Figure 8. Meshes with

6 \times 6

and

18 \times 18

elements.

Table 1. Shape regularity parameters for each mesh

T_{h}^{2}

Table 1. Shape regularity parameters for each mesh

T_{h}^{2}

	$n = 6$	$n = 10$	$n = 14$	$n = 18$	$n = 22$
$σ_{T_{h}}$	0.180	0.115	0.161	0.127	0.150

Table 2.

L^{2}

errors and convergence rates for

{DS}_{r}^{R}

and

{DS}_{r}^{C}

Table 2.

L^{2}

errors and convergence rates for

{DS}_{r}^{R}

and

{DS}_{r}^{C}

	$r = 2$				$r = 3$
	${DS}_{2}^{R}$		${DS}_{2}^{C}$		${DS}_{3}^{R}$		${DS}_{3}^{C}$
n	error	rate	error	rate	error	rate	error	rate
10	2.160e-04	3.45	2.144e-04	3.50	8.859e-06	4.34	1.031e-05	4.35
14	7.329e-05	3.16	7.165e-05	3.21	2.175e-06	4.11	2.518e-06	4.13
18	3.452e-05	2.95	3.409e-05	2.92	7.927e-07	3.96	8.964e-07	4.05
22	1.863e-05	3.47	1.841e-05	3.46	3.555e-07	4.51	4.045e-07	4.48
	$r = 4$				$r = 5$
	${DS}_{4}^{R}$		${DS}_{4}^{C}$		${DS}_{5}^{R}$		${DS}_{5}^{C}$
n	error	rate	error	rate	error	rate	error	rate
10	3.467e-07	5.69	3.972e-07	6.11	1.133e-08	6.97	1.730e-08	6.61
14	5.644e-08	5.31	6.622e-08	5.24	1.202e-09	6.57	1.964e-09	6.37
18	1.530e-08	5.12	1.823e-08	5.06	4.376e-10	3.97	4.134e-10	6.12
22	5.314e-09	5.95	6.239e-09	6.03	8.905e-11	8.95	1.243e-10	6.76

Table 3.

H^{1}

-seminorm errors and convergence rates for

{DS}_{r}^{R}

and

{DS}_{r}^{C}

Table 3.

H^{1}

-seminorm errors and convergence rates for

{DS}_{r}^{R}

and

{DS}_{r}^{C}

	$r = 2$				$r = 3$
	${DS}_{2}^{R}$		${DS}_{2}^{C}$		${DS}_{3}^{R}$		${DS}_{3}^{C}$
n	error	rate	error	rate	error	rate	error	rate
10	3.561e-03	2.32	3.552e-03	2.36	1.933e-04	3.13	2.390e-04	3.11
14	1.683e-03	2.19	1.660e-03	2.23	6.724e-05	3.09	8.343e-05	3.08
18	1.018e-03	1.97	1.013e-03	1.94	3.144e-05	2.98	3.783e-05	3.10
22	6.712e-04	2.34	6.696e-04	2.33	1.730e-05	3.36	2.114e-05	3.27
	$r = 4$				$r = 5$
	${DS}_{4}^{R}$		${DS}_{4}^{C}$		${DS}_{5}^{R}$		${DS}_{5}^{C}$
n	error	rate	error	rate	error	rate	error	rate
10	8.530e-06	4.55	1.027e-05	4.91	3.103e-07	5.73	4.394e-07	5.57
14	1.973e-06	4.29	2.439e-06	4.21	4.625e-08	5.57	7.098e-08	5.34
18	6.952e-07	4.09	8.785e-07	4.01	2.646e-08	2.19	1.981e-08	5.01
22	2.969e-07	4.78	3.689e-07	4.88	5.973e-09	8.37	7.233e-09	5.66

Table 4. Errors and convergence rates at

n = 18

computed from the previous step

n = 14

, for the direct serendipity spaces

{DS}_{r}^{R}

{DS}_{r}^{V}

{DS}_{r}^{C}

{DS}_{r}^{M}

, and

{DS}_{r}^{T}

Table 4. Errors and convergence rates at

n = 18

computed from the previous step

n = 14

, for the direct serendipity spaces

{DS}_{r}^{R}

{DS}_{r}^{V}

{DS}_{r}^{C}

{DS}_{r}^{M}

, and

{DS}_{r}^{T}

	$r = 2$		$r = 3$		$r = 4$		$r = 5$
	error	rate	error	rate	error	rate	error	rate
$L^{2}$ errors and convergence rates
${DS}_{r}^{R}$	3.452e-05	2.95	7.927e-07	3.96	1.530e-08	5.12	4.376e-10	3.97
${DS}_{r}^{V}$	3.554e-05	2.89	1.073e-06	3.87	2.108e-08	4.83	4.637e-10	5.92
${DS}_{r}^{C}$	3.409e-05	2.92	8.964e-07	4.05	1.823e-08	5.06	4.134e-10	6.12
${DS}_{r}^{M}$	6.820e-05	2.88	1.697e-06	3.85	3.095e-08	4.80	6.115e-10	6.02
${DS}_{r}^{T}$	7.072e-05	2.92	1.866e-06	4.04	3.367e-08	4.91	5.830e-10	6.07
$H^{1}$ -seminorm errors and convergence rates
${DS}_{r}^{R}$	1.018e-03	1.97	3.144e-05	2.98	6.952e-07	4.09	2.646e-08	2.19
${DS}_{r}^{V}$	1.059e-03	1.89	4.199e-05	2.93	9.959e-07	3.85	2.184e-08	4.80
${DS}_{r}^{C}$	1.013e-03	1.94	3.783e-05	3.10	8.785e-07	4.01	1.981e-08	5.01
${DS}_{r}^{M}$	1.976e-03	1.88	6.334e-05	3.01	1.590e-06	3.95	3.076e-08	4.83
${DS}_{r}^{T}$	2.059e-03	1.94	6.895e-05	3.16	1.710e-06	4.04	3.008e-08	4.87

Table 5. Errors and convergence rates at

n = 18

computed from the previous step

n = 14

, for the reduced

H (div)

-approximation spaces

V_{r}^{R, r - 1}

V_{r}^{V, r - 1}

V_{r}^{C, r - 1}

V_{r}^{M, r - 1}

, and

V_{r}^{T, r - 1}

Table 5. Errors and convergence rates at

n = 18

computed from the previous step

n = 14

, for the reduced

H (div)

-approximation spaces

V_{r}^{R, r - 1}

V_{r}^{V, r - 1}

V_{r}^{C, r - 1}

V_{r}^{M, r - 1}

, and

V_{r}^{T, r - 1}

	$∥ p - p_{h} ∥$		$∥ u - u_{h} ∥$		$∥ \nabla \cdot (u - u_{h}) ∥$
	error	rate	error	rate	error	rate
$r = 1$ , reduced $H (div)$ -approximation
$V_{1}^{R, 0}$	7.039e-02	1.01	5.428e-03	1.98	6.988e-02	0.99
$V_{1}^{V, 0}$	7.039e-02	1.01	5.429e-03	1.98	6.988e-02	0.99
$V_{1}^{C, 0}$	7.039e-02	1.01	5.443e-03	1.98	6.988e-02	0.99
$V_{1}^{M, 0}$	7.039e-02	1.01	5.366e-03	1.98	6.988e-02	0.99
$V_{1}^{T, 0}$	7.039e-02	1.01	5.362e-03	1.98	6.988e-02	0.99
$r = 2$ , reduced $H (div)$ -approximation
$V_{2}^{R, 1}$	2.614e-03	1.96	8.492e-05	2.92	2.614e-03	1.96
$V_{2}^{V, 1}$	2.614e-03	1.96	8.876e-05	2.89	2.614e-03	1.96
$V_{2}^{C, 1}$	2.614e-03	1.96	8.850e-05	2.87	2.614e-03	1.96
$V_{2}^{M, 1}$	2.614e-03	1.96	8.895e-05	2.85	2.614e-03	1.96
$V_{2}^{T, 1}$	2.614e-03	1.96	8.973e-05	2.85	2.614e-03	1.96
$r = 3$ , reduced $H (div)$ -approximation
$V_{3}^{R, 2}$	6.515e-05	2.96	1.887e-06	3.90	6.515e-05	2.96
$V_{3}^{V, 2}$	6.515e-05	2.96	1.931e-06	3.89	6.515e-05	2.96
$V_{3}^{C, 2}$	6.515e-05	2.96	1.911e-06	3.89	6.515e-05	2.96
$V_{3}^{M, 2}$	6.515e-05	2.96	2.007e-06	3.81	6.515e-05	2.96
$V_{3}^{T, 2}$	6.515e-05	2.96	2.105e-06	3.83	6.515e-05	2.96

Table 6. Errors and convergence rates at

n = 18

computed from the previous step

n = 14

, for the full

H (div)

-approximation spaces

V_{r}^{R, r}

V_{r}^{V, r}

V_{r}^{C, r}

V_{r}^{M, r}

, and

V_{r}^{T, r}

Table 6. Errors and convergence rates at

n = 18

computed from the previous step

n = 14

, for the full

H (div)

-approximation spaces

V_{r}^{R, r}

V_{r}^{V, r}

V_{r}^{C, r}

V_{r}^{M, r}

, and

V_{r}^{T, r}

	$∥ p - p_{h} ∥$		$∥ u - u_{h} ∥$		$∥ \nabla \cdot (u - u_{h}) ∥$
n	error	rate	error	rate	error	rate
$r = 0$ , full $H (div)$ -approximation
$V_{0}^{R, 0}$	7.030e-02	1.01	2.701e-02	1.10	6.988e-02	0.99
$V_{0}^{V, 0}$	7.027e-02	1.01	3.095e-02	1.03	6.988e-02	0.99
$V_{0}^{C, 0}$	7.028e-02	1.01	2.951e-02	1.03	6.988e-02	0.99
$V_{0}^{M, 0}$	7.027e-02	1.01	3.065e-02	0.93	6.988e-02	0.99
$V_{0}^{T, 0}$	7.026e-02	1.01	3.163e-02	0.92	6.988e-02	0.99
$r = 1$ , full $H (div)$ -approximation
$V_{1}^{R, 1}$	2.614e-03	1.96	4.895e-04	2.19	2.614e-03	1.96
$V_{1}^{V, 1}$	2.614e-03	1.96	5.542e-04	2.13	2.614e-03	1.96
$V_{1}^{C, 1}$	2.614e-03	1.96	5.226e-04	2.17	2.614e-03	1.96
$V_{1}^{M, 1}$	2.614e-03	1.96	7.505e-04	2.08	2.614e-03	1.96
$V_{1}^{T, 1}$	2.614e-03	1.96	7.917e-04	2.15	2.614e-03	1.96
$r = 2$ , full $H (div)$ -approximation
$V_{2}^{R, 2}$	6.515e-05	2.96	8.818e-06	3.10	6.515e-05	2.96
$V_{2}^{V, 2}$	6.515e-05	2.96	1.887e-05	2.92	6.515e-05	2.96
$V_{2}^{C, 2}$	6.515e-05	2.96	1.526e-05	3.03	6.515e-05	2.96
$V_{2}^{M, 2}$	6.515e-05	2.96	2.801e-05	2.49	6.515e-05	2.96
$V_{2}^{T, 2}$	6.515e-05	2.96	3.010e-05	2.67	6.515e-05	2.96
$r = 3$ , full $H (div)$ -approximation
$V_{3}^{R, 3}$	1.182e-06	3.99	2.144e-07	3.65	1.182e-06	3.99
$V_{3}^{V, 3}$	1.182e-06	3.99	3.324e-07	3.43	1.182e-06	3.99
$V_{3}^{C, 3}$	1.182e-06	3.99	2.933e-07	3.50	1.182e-06	3.99
$V_{3}^{M, 3}$	1.183e-06	3.99	1.254e-06	3.10	1.182e-06	3.99
$V_{3}^{T, 3}$	1.183e-06	3.99	1.547e-06	3.61	1.182e-06	3.99

Table 7. Errors and convergence rates in

L^{2}

for

V_{r}^{C, r - 1}

Table 7. Errors and convergence rates in

L^{2}

for

V_{r}^{C, r - 1}

	$∥ p - p_{h} ∥$		$∥ u - u_{h} ∥$		$∥ \nabla \cdot (u - u_{h}) ∥$
n	error	rate	error	rate	error	rate
$r = 1$ , reduced $H (div)$ -approximation
10	1.290e-01	1.24	1.775e-02	2.29	1.260e-01	1.15
14	9.109e-02	1.02	9.024e-03	1.98	9.001e-02	0.98
18	7.039e-02	1.01	5.443e-03	1.98	6.988e-02	0.99
22	5.736e-02	1.15	3.630e-03	2.28	5.708e-02	1.14
$r = 2$ , reduced $H (div)$ -approximation
10	8.635e-03	2.23	5.210e-04	3.28	8.634e-03	2.23
14	4.308e-03	2.04	1.841e-04	3.04	4.308e-03	2.03
18	2.614e-03	1.96	8.850e-05	2.87	2.614e-03	1.96
22	1.715e-03	2.37	4.772e-05	3.47	1.715e-03	2.37
$r = 3$ , reduced $H (div)$ -approximation
10	3.881e-04	3.38	2.021e-05	4.39	3.881e-04	3.38
14	1.384e-04	3.02	5.151e-06	4.00	1.384e-04	3.02
18	6.515e-05	2.96	1.911e-06	3.89	6.515e-05	2.96
22	3.507e-05	3.48	8.432e-07	4.60	3.507e-05	3.48

Table 8. Errors and convergence rates in

L^{2}

for

V_{r}^{C, r}

Table 8. Errors and convergence rates in

L^{2}

for

V_{r}^{C, r}

	$∥ p - p_{h} ∥$		$∥ u - u_{h} ∥$		$∥ \nabla \cdot (u - u_{h}) ∥$
n	error	rate	error	rate	error	rate
$r = 0$ , full $H (div)$ -approximation
10	1.281e-01	1.20	6.389e-02	1.54	1.260e-01	1.15
14	9.086e-02	1.01	3.832e-02	1.50	9.001e-02	0.98
18	7.028e-02	1.01	2.951e-02	1.03	6.988e-02	0.99
22	5.731e-02	1.15	2.145e-02	1.79	5.708e-02	1.14
$r = 1$ , full $H (div)$ -approximation
10	8.635e-03	2.23	2.003e-03	2.65	8.634e-03	2.23
14	4.308e-03	2.04	9.076e-04	2.32	4.308e-03	2.03
18	2.614e-03	1.96	5.226e-04	2.17	2.614e-03	1.96
22	1.715e-03	2.37	3.320e-04	2.55	1.715e-03	2.37
$r = 2$ , full $H (div)$ -approximation
10	3.881e-04	3.38	1.007e-04	3.47	3.881e-04	3.38
14	1.384e-04	3.02	3.303e-05	3.26	1.384e-04	3.02
18	6.515e-05	2.96	1.526e-05	3.03	6.515e-05	2.96
22	3.507e-05	3.48	7.889e-06	3.71	3.507e-05	3.48
$r = 3$ , full $H (div)$ -approximation
10	1.294e-05	4.60	3.537e-06	4.84	1.294e-05	4.60
14	3.270e-06	4.03	7.157e-07	4.68	3.270e-06	4.03
18	1.182e-06	3.99	2.933e-07	3.50	1.182e-06	3.99
22	5.219e-07	4.60	1.301e-07	4.57	5.219e-07	4.60

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Construction of Supplemental Functions for Direct Serendipity and Mixed Finite Elements on Polygons

Abstract

1. Introduction

2. Notation

3. Direct serendipity and mixed finite elements

3.1. Direct serendipity finite elements

3.2. Direct mixed finite elements

4. Piecewise continuous supplements

4.1. Supplemental functions in H 1 ( E N )

4.2. Supplemental functions in H p ( E N )

5. Approximation properties

6. Numerical results

6.1. The meshes used

6.2. Results for direct serendipity spaces

6.3. Results for direct mixed spaces

7. Conclusions

Author Contributions

Funding

Data Availability Statement

Conflicts of Interest

References

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4.1. Supplemental functions in $H^{1} (E_{N})$

4.2. Supplemental functions in $H^{p} (E_{N})$