Convergence of Lagrange finite elements for the Maxwell Eigenvalue Problem in 2D

03/18/2020 ∙ by Daniele Boffi, et al. ∙ King Abdullah University of Science and Technology Brown University University of Pittsburgh 0

We consider finite element approximations of the Maxwell eigenvalue problem in two dimensions. We prove, in certain settings, convergence of the discrete eigenvalues using Lagrange finite elements. In particular, we prove convergence in three scenarios: piecewise linear elements on Powell–Sabin triangulations, piecewise quadratic elements on Clough–Tocher triangulations, and piecewise quartics (and higher) elements on general shape-regular triangulations. We provide numerical experiments that support the theoretical results. The computations also show that, on general triangulations, the eigenvalue approximations are very sensitive to nearly singular vertices, i.e., vertices that fall on exactly two "almost" straight lines.

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1. Introduction

Let be a contractible polygonal domain and consider the eigenvalue problem

(1.1)

where , , and denotes the inner product over . Given a finite element space , a finite element method seeks and satisfying

(1.2)

For example, one can take to be the -conforming Nedelec finite elements (i.e., the rotated Raviart-Thomas finite elements) as the finite element space. It is well-known this choice leads to a convergent approximation of the eigenvalue problem. On the other hand, taking as a space continuous piecewise polynomials (i.e., a -conforming Lagrange finite element) may lead to spurious eigenvalues for any mesh parameter.

There is a vast literature on this subject. The interested reader is referred to [5, Section 20] for an extensive survey including a comprehensive list of references about Nedelec finite elements and to [8, 6] for a discussion about the use of standard Lagrange finite elements (see also [3] for a discussion of these phenomena in the context of the finite element exterior calculus).

To better appreciate the problem and its discretization, we consider the equivalent formulation introduced in [8] for :

(1.3a)
(1.3b)

Taking with shows the equivalence of (1.3) and (1.1) with , and .

The corresponding finite element method for the mixed formulation (1.3) seeks , , and such that

(1.4a)
(1.4b)

with . Similar to the continuous problem, if the finite element spaces satisfy , then the mixed finite element formulation (1.4) is equivalent to the primal one (1.2) with and .

If is the Nedelec space of index , then we may take to be the space of piecewise polynomials of degree . In this case, forms an inf-sup stable pair of spaces, in particular, there exists a Fortin projection

satisfying

(1.5a)
(1.5b)

Here . Moreover, is a parameter such that [2], and is the orthogonal projection onto . Using this projection one can prove that the corresponding source problems converges uniformly, and this is sufficient to prove convergence of the eigenvalue problem (1.2) (see [18, 5] and Proposition 2.1).

On the other hand, if is taken to be the Lagrange finite element space of degree , then a natural choice of is the space of (discontinuous) piecewise polynomials of degree . However, is not inf-sup stable on generic triangulations, at least when [21, 7], and therefore there does not exist a Fortin projection satisfying (1.5). On the other hand, the pair is known to be stable on special triangulations, even if the inf-sup condition might not be sufficient to guarantee the existence of a Fortin projector satisfying (1.5) (see [6]). On very special triangulations, Wong and Cendes [24] showed numerically that solutions to (1.2) do converge to the correct eigenvalues using piecewise linear Lagrange elements (i.e., ). In fact, they used precisely the Powell–Sabin triangulations (see Figure 1). In this paper, we prove that indeed using Lagrange elements in conjunction with Powell–Sabin triangulation leads to a convergent method. We do this by proving that there is a Fortin projection of sorts. We show that there exists an operator satisfying

(1.6a)
(1.6b)

where . Note that (1.5) implies (1.6), and we prove convergence of the eigenvalue problem whenever there is a projection satisfying (1.6). In addition to linear Lagrange elements on Powell–Sabin triangulations, we prove the existence of such a projection on Clough–Tocher splits using quadratic Lagrange elements, and on general triangulations using th degree Lagrange elements with (i.e., the Scott–Vogelius finite elements). For the Scott–Vogelius finite elements, we find the approximate eigenvalues are extremely sensitive if the mesh has nearly singular vertices, i.e., vertices that fall on exactly two “almost” straight lines (cf. Section 3.3). We give numerical examples that illustrate this behavior.

Recently Duan et al. [12] considered Lagrange finite elements for Maxwell’s eigenvalue problem in two and three dimensions. However, they use a different formulation than ours and they also add a stabilization term.

The paper is organized as follows: In the next section we give a convergence proof for finite elements spaces with stable projections. In Section 3, we provide three examples of Lagrange finite element spaces with stable projections: the piecewise linear Lagrange space on Powell–Sabin splits, the piecewise quadratic Lagrange space on Clough–Tocher splits, and the piecewise th degree Lagrange space on generic triangulations. Finally, in Section 4 we provide numerical experiments.

2. Convergence Framework

Define the two-dimensional , , and divergence operators as

and define the Hilbert spaces

where

is a unit tangent vector of

. Recall that .

Let and be finite element spaces such that .

2.1. Source problems

We will require the corresponding source problems for the analysis. To this end, we define the solution operators and such that for given , there holds

(2.1a)
(2.1b)

Likewise, the discrete source problem is given by:

Find and such that
(2.2a)
(2.2b)

Note that , and so . Moreover, using that we have that .

We define the operator norm:

(2.3)

We will use the next standard result that states that the uniform convergence of the discrete source problem implies convergence of the discrete eigenvalues. This result is a consequence of the classical discussion in [4, Section 8] (see also [5, Section 9] and [8, Theorem 4.4]).

Proposition 2.1.

Let and be defined from (2.1) and (2.2), respectively, and suppose that as . Consider the problem (1.3) and consider the nonzero eigenvalues . Consider also (1.4) and its non-zero eigenvalues . Then, for any fixed , .

Therefore, to prove convergence of eigenvalues it suffices to show uniform convergence of the discrete source problem. To prove this, we will exploit the embedding along with an assumption on the finite element spaces. The embedding result is proved in three dimensions in [2], and we state the two dimensional version here.

Proposition 2.2.

Let be a contractible polygonal domain. Then there exists constants and such that

From now on will refer to the delta of the above proposition. We will use the following space

(2.4)
Assumption 2.3.

We assume that and the existence of a projection such that

(2.5a)
(2.5b)

Furthermore, we assume that the -orthogonal projection satisfies

Here, the constants are assumed to satisfy and for .

Theorem 2.4.

Suppose that satisfy Assumption 2.3. Let and be defined by (2.1) and (2.2), respectively. Then there holds

Note that Theorem 2.4 and Proposition 2.1 imply that the discrete eigenvalues in the finite element method (1.2) converge to the correct values. To prove Theorem 2.4, we require two preliminary results.

Lemma 2.5.

Suppose that Assumption 2.3 is satisfied. Then there exists a constant such that

Proof.

Let and set , and . We see that

(2.6a)
(2.6b)

Setting in (2.6b) and in (2.6a), and adding the result yields . Furthermore, (2.6a) implies . Therefore, there holds

However, the properties of the projection and Assumption 2.3 give us

Thus, we have shown

Finally, by the Poincare’s inequality we have

Next we prove that Assumption 2.3 implies the inf-sup condition for the pair .

Lemma 2.6.

Suppose that Assumption 2.3 is satisfied. Then there exists a constant such that for every , there exists such that and .

Proof.

Let with such that . Noting that , we define so that . Moreover,

Now we can prove Theorem 2.4.

Proof of Theorem 2.4.

Let , and set and . Let and .

We first derive an estimate for

. Using the inclusion , we see that

Setting and applying the Cauchy–Schwarz inequality yields

If we use Proposition 2.2 we get

Hence,

Next, we note that by Lemma 2.5,

and therefore,

Using the inf-sup stability stated in Lemma 2.6, we have

Hence, we have

But we have , and so

3. Examples of Fortin Operators

In this section we give examples of finite element pairs satisfying Assumption 2.3, where is taken to be a space of continuous, piecewise polynomials, i.e., a Lagrange finite element space. Here we use recent results on divergence-free finite element pairs for the Stokes problem to construct a Fortin projection satisfying (2.5). A common theme of these Stokes pairs is the imposition of mesh conditions for low-polynomial degree finite element spaces; it is well-known that Assumption 2.3 is not satisfied on general simplicial meshes and for low polynomial degree. Before continuing, we introduce some notation.

We denote by a shape-regular, simplicial triangulation of with for all , and . Let , , denote the sets of interior vertices, boundary vertices, and corner vertices, respectively. Note that the cardinality of is uniformly bounded due to the shape-regularity of . The set of all vertices is . Likewise, and are the sets of interior and boundary edges, respectively, and . We denote by the patch of triangles that have as a vertex. Likewise, and are the sets of interior and boundary vertices of , and is the set of interior edges of .

For a non-negative integer and set , let to be the space of piecewise polynomials of degree with domain . The analogous space of piecewise polynomials with respect to is

and the Lagrange finite element space is

Analogous vector-valued spaces are denoted in boldface, e.g., . Finally, the constant denotes a generic constant that is independent of the mesh parameter and may take different values at different occurrences.

In the subsequent sections, we will employ a Scott–Zhang type interpolant on the space

. We cannot use the Scott–Zhang interpolant directly, as the canonical Scott–Zhang interpolant of a function in might not have zero tangential components at the corners of ; hence, we have to modify the Scott–Zhang interpolant at the corners of . We give the detailed construction in the appendix but we state the result here.

Lemma 3.1.

There exists an interpolant with the following bound: If does not have a corner vertex (i.e., ), then

(3.1)

where . Otherwise, if for some , then

(3.2)

3.1. Construction of Fortin Operator on Powell–Sabin Splits

In this section, we use the recent results given in [16] to construct a Fortin projection into the Lagrange finite element space defined on Powell–Sabin triangulations. For simplicity and readability, we focus on the lowest-order case; however, the arguments easily extend to arbitrary polynomial degree .

Given the simplicial triangulation of of , we construct its Powell–Sabin refinement as follows [20, 19, 16]: First, adjoin the incenter of each to each vertex of . Next, the interior points (incenters) of each adjacent pair of triangles are connected with an edge. For any that shares an edge with the boundary of , the midpoint of that edge is connected with the incenter of . Thus, each is split into six triangles; cf. Figure 1.

Figure 1. A simplicial triangulation of the unit square (left) and the associated Powell–Sabin triangulation (right).

Let be the points of intersection of the interior edges of that adjoin incenters, let be the intersection points of the boundary edges that adjoin incenters, and set . Note that, by the definition of the Powell–Sabin split, the points in are the singular vertices in , i.e, the vertices that lie on exactly two straight lines. In particular, for a vertex there exists four triangles such that is a vertex of . Without loss of generality we assume that these triangles are labeled in a counterclockwise direction. We then define for a scalar function ,

(3.3)

We then define the spaces

(3.4a)
(3.4b)
Lemma 3.2 ([16]).

Let and be defined by (3.4). Then there holds .

We now extend the results of [16] to construct an appropriate Fortin operator that is well defined for . To do so, we require some additional notation.

For an interior singular vertex , let be a triangle in such that , and let be the triangles in such that and . Let , and let be the outward unit normal of perpendicular to . We then define the jump of a scalar piecewise smooth function at (restricted to ) as

Note that is single-valued for all . In particular, if is an interior singular vertex with for some , , then for all because . Therefore, we shall omit the subscript and simply write .

Next for a triangle in the non-refined mesh, we denote by the resulting set of three triangles obtained by connecting the barycenter of to its vertices, i.e., is the Clough–Tocher refinement of . We define the set of (local) piecewise polynomials with respect to this partition as

(3.5)

The following lemma provides the degrees of freedom for

and that will be used to construct the Fortin operator. The result essentially follows from [16, Lemma 4.5].

Lemma 3.3.

A function is uniquely defined by the conditions

(3.6a)
(3.6b)
(3.6c)
(3.6d)
(3.6e)

Moreover, a function is uniquely determined by the values

(3.7a)
(3.7b)
Theorem 3.4.

Let and be defined by (3.4), and let be defined by (2.4). Then there exists a projection such that for all . Moreover,