Discontinuous Petrov–Galerkin Approximation of Eigenvalue Problems

Fleurianne Bertrand; Daniele Boffi; Henrik Schneider

doi:10.1515/cmam-2022-0069

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Discontinuous Petrov–Galerkin Approximation of Eigenvalue Problems

Fleurianne Bertrand , Daniele Boffi and Henrik Schneider

Published/Copyright: May 26, 2022

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From the journal Computational Methods in Applied Mathematics Volume 23 Issue 1

Abstract

In this paper, the discontinuous Petrov–Galerkin approximation of the Laplace eigenvalue problem is discussed. We consider in particular the primal and ultraweak formulations of the problem and prove the convergence together with a priori error estimates. Moreover, we propose two possible error estimators and perform the corresponding a posteriori error analysis. The theoretical results are confirmed numerically, and it is shown that the error estimators can be used to design an optimally convergent adaptive scheme.

Keywords: Eigenvalue Problems; DPG Methods

MSC 2010: 65N25

1 Introduction

DPG approximation of partial differential equations is a popular and effective technique which has reached a quite mature level of discussion within the scientific community. The literature about DPG is pretty rich. The method has been introduced in a series of papers [16, 17, 20, 30] during the last decade. Originally, the method has been presented as a technique to design an intrinsically stabilized scheme for advective problems. The main idea is to use suitable discontinuous trial and test functions that are tailored for stability. The ideal DPG formulation is turned into a practical DPG formulation [23] where the test function space is easily computable and arbitrarily close to the optimal one. The DPG formulation comes with a natural a posteriori error indicator that can be used for driving a robust h ⁢ p adaptivity. Moreover, it has been shown that, for one-dimensional problems, the DPG method can provide no phase errors in the case of time-harmonic wave propagation.

After these pioneer works, several studies have been performed, showing that the method can be applied to a variety of other problems. In particular, a solid analysis has been presented in the case of the Laplace equations [18]; the method has been proved to be locking free for linear elasticity [6]; it has been applied to Friedrichs-like systems [8], including convection-diffusion-reaction, linear continuum mechanics, time-domain acoustic, and a version of Maxwell’s equations; it has been applied to the Reissner–Mindlin plate bending model [9], to the Helmholtz equation [22], to the Stokes problem [26], to compressible flows [15], to the Navier–Stokes equations [27], and to the Maxwell equation [11].

In this paper, we are interested in the approximation of the Laplace eigenvalue problem. We study the so-called primal and ultraweak formulation of the Poisson problem: we refer the interested reader to [19, 18, 23] for the analysis and a discussion of the primal and ultraweak formulations for the Laplace problem in the case of the source problem. We will also look at a posteriori error estimators, perform an a posteriori analysis, and show numerically the optimal convergence of an adaptive scheme.

Following what was recently done for the least-squares finite element method [3, 2], we study how the DPG method can approximate eigenvalue problems.

After recalling the abstract setting for DPG approximations and the convergence of eigenvalues and eigenfunctions in Section 2, we apply the theoretical framework to the Laplace eigenvalue problem in Section 3, where we show the a priori estimates for the primal and ultraweak formulations. The a posteriori analysis is developed in Section 4 where we consider a natural error estimator related to what was studied in [10] and the alternative error estimator introduced in [24] which is based on a suitable characterization of the eigenfunctions in terms of Crouzeix–Raviart elements valid for the lowest order approximation. Finally, in Section 5, we present some numerical tests where the theory is confirmed and where it is shown that an adaptive scheme for the DPG approximation of the eigensolutions of the Laplace problem is optimally convergent.

2 Problem Formulation and A Priori Analysis Framework

We start by describing the general structure of a DPG source problem, since this is useful in order to introduce our notation before dealing with the corresponding eigenvalue problem.

The source problem we are studying is: find u ∈ U such that

(2.1) b ⁢ ( u , v ) = ℓ ⁢ ( v ) for all ⁢ v ∈ V ,

where 𝑈 is a trial Hilbert space and 𝑉 is a test Hilbert space. The bilinear form b : U × V → C satisfies the assumptions

(2.2) b ⁢ ( u , v ) = 0 ⁢ for all ⁢ u ∈ V ⟹ v = 0 ,

(2.3) C 1 ⁢ ∥ w ∥ U ≤ sup v ∈ V ∖ { 0 } ⁡ | b ⁢ ( w , v ) | ∥ v ∥ V ≤ C 2 ⁢ ∥ w ∥ U for all ⁢ v ∈ V ,

ℓ : V → C is a linear form.

The DPG formulation of (2.1) is obtained by introducing a discrete space U h ⊂ U and an optimal test function space given by V opt = T ⁢ ( U h ) , where T : U → V is the trial-to-test operator defined as: find T ⁢ u ∈ V such that

(2.4) ( T ⁢ u , v ) V = b ⁢ ( u , v ) for all ⁢ v ∈ V .

The discrete source problem is: find u h ∈ U h such that

b ⁢ ( u h , v ) = ℓ ⁢ ( v ) for all ⁢ v ∈ V opt .

This is an ideal setting and the stated assumptions (2.2), (2.3) are sufficient to prove well-posedness, but the actual computation of the test function space V opt is not feasible in most applications. So, in general, a practical DPG method is adopted where the test space V opt is replaced by V opt , h , obtained after introducing a finite-dimensional subspace V h ⊂ V and defining a discrete trial-to-test operator T h as in (2.4) with 𝑉 replaced by V h . Then the practical optimal test space is V opt , h = T h ⁢ ( U h ) . If V h is a space of discontinuous piecewise polynomials, then the computation of the test functions is cheap, involving the solution of a block diagonal system.

We assume that there exist a linear operator Π : V → V h and C Π such that, for all u h ∈ U h and all v ∈ V ,

b ⁢ ( u h , v - Π ⁢ v ) = 0 ,

(2.5) ∥ Π ⁢ v ∥ V ≤ C Π ⁢ ∥ v ∥ V .

A fundamental characterization of the solution u h of the DPG system is given by the following mixed problem that is defined only via the discrete spaces U h and V h without the need of the trial-to-test operator 𝑇: find u h ∈ U h and ε h ∈ V h such that

(2.6) { ( ε h , v h ) V + b ⁢ ( u h , v h ) = ℓ ⁢ ( v h ) for all ⁢ v h ∈ V h , b ⁢ ( z h , ε h ) ¯ = 0 for all ⁢ z h ∈ U h .

We are now ready to introduce the eigenvalue problem associated with (2.1) and its approximation corresponding to (2.6).

Usually, the space 𝑈 consists of two components and can be presented as U = U 0 × U 1 , where U 0 is a functional space defined on Ω (volumetric part) and U 1 is the remaining part that can be defined on Ω or on the skeleton of a given triangulation. Let ℋ be a Hilbert pivot space so that we have the usual triplet U 0 ⊂ H ≃ H ′ ⊂ U 0 ′ , and consider a bilinear form m : H × V → C . The continuous eigenvalue problem is: find eigenvalues λ ∈ C and eigenfunctions u = ( u 0 , u 1 ) ∈ U = U 0 × U 1 with u 0 ≠ 0 such that

(2.7) b ⁢ ( u , v ) = λ ⁢ m ⁢ ( u 0 , v ) for all ⁢ v ∈ V .

In order to state the appropriate (compactness) assumptions, we introduce the solution operator T F : H → H as follows: T F ⁢ f ∈ H is the component u 0 of the solution u ∈ U to

(2.8) b ⁢ ( u , v ) = m ⁢ ( f , v ) for all ⁢ v ∈ V .

We assume that (2.8) is uniquely solvable and the operator T F is compact.

The discrete space U h is analogously made of two components U 0 , h ⊂ U 0 and U 1 , h ⊂ U 1 . The DPG discretization, corresponding to the mixed formulation (2.6), is given by: find λ h ∈ C such that, for some u h = ( u 0 , h , u 1 , h ) ∈ U h = U 0 , h × U 1 , h with u 0 , h ≠ 0 and some ε h ∈ V h , it holds

(2.9) { ( ε h , v h ) V + b ⁢ ( u h , v h ) = λ h ⁢ m ⁢ ( u 0 , h , v h ) for all ⁢ v h ∈ V h , b ⁢ ( z h , ε h ) ¯ = 0 for all ⁢ z h ∈ U h .

The matrix form of this formulation is given by

( A B ⊤ B 0 ) ⁢ ( x y ) = λ ⁢ ( 0 M 0 0 ) ⁢ ( x y ) ,

where 𝗑 is the vector representation of ε h ∈ V h and 𝗒 of u h ∈ U h . We introduce the discrete counterpart T F , h : H → H of the solution operator T F as follows: T F , h ⁢ f ∈ U 0 , h ⊂ H is the component u 0 , h ∈ U 0 , h of the solution u h ∈ U of the following problem, for some ε h ∈ V h :

(2.10) { ( ε h , v h ) V + b ⁢ ( u h , v h ) = m ⁢ ( f , v h ) for all ⁢ v h ∈ V h , b ⁢ ( z h , ε h ) ¯ = 0 for all ⁢ z h ∈ U h .

In order show a priori estimates for the eigensolution computed with the DPG method, we are going to use the classical Babǔska–Osborn theory (see [1, Section 8] and [4, Section 9]). Let us denote by λ i , i = 1 , … , the eigenvalues of the continuous problem (2.7) sorted such that 0 < | λ 1 | ≤ | λ 2 | ≤ ⋯ and by E i = span ⁢ ( u 0 , i ) , i = 1 , … , the corresponding eigenspaces. In case of multiple eigenvalues, we repeat them so that each E i is one-dimensional. Analogous notation λ i , h and E i , h , i = 1 , … , is adopted for the discrete problem (2.9).

Theorem 1

If the following convergence in norm holds true:

(2.11) ∥ ( T F - T F , h ) ⁢ f ∥ H ≤ ρ ⁢ ( h ) ⁢ ∥ f ∥ H for all ⁢ f ∈ H

with ρ ⁢ ( h ) converging to zero as ℎ goes to zero, then the discrete eigenvalues and eigenfunctions converge to the continuous ones. That is, any compact set 𝐾 included in the resolvent set of T F is included in the resolvent set of T F , h for ℎ small enough (absence of spurious modes); moreover, if λ i is an eigenvalue of algebraic multiplicity 𝑚, then there are exactly 𝑚 discrete eigenvalues λ i j , h , j = 1 , … , m , converging to λ i (convergence).

In order to estimate the rate of convergence, as usual, we shall make use of the gap between subspaces of Hilbert spaces defined as

δ ^ ⁢ ( A , B ) = max ⁡ { δ ⁢ ( A , B ) , δ ⁢ ( B , A ) } ,

where

δ ⁢ ( A , B ) = sup a ∈ A ∥ a ∥ = 1 ⁡ δ ⁢ ( a , B ) with ⁢ δ ⁢ ( a , B ) = inf b ∈ B ⁡ ∥ a - b ∥ W

for 𝐴 and 𝐵 closed subspaces of a Hilbert space 𝑊. If W = U 0 and 𝐸 is the 𝑚-dimensional eigenspace of the continuous problem corresponding to λ i (see the setting of Theorem 1), we introduce the following quantity related to 𝐸:

γ h = ∥ ( T F - T F , h ) | E ∥ L ⁢ ( U 0 ) ,

and if E ′ is the corresponding eigenspace of the adjoint operator T F * , we consider the following quantity:

γ h * = ∥ ( T F * - T F , h * ) | E ′ ∥ L ⁢ ( U 0 ′ ) ,

where T F , h * is the discrete solution operator associated with the adjoint problem. Then we recall the following classical result.

Theorem 2

Under the hypothesis of Theorem 1, there exists a constant C > 0 such that δ ^ ⁢ ( E h , E ) ≤ C ⁢ γ h , where E h is the direct sum of the eigenspaces of the 𝑚 eigenvalues approximating λ i . Moreover, if 𝛼 is the ascent multiplicity of λ i , there exists a constant C > 0 so that max j = 1 , … , m ⁡ | λ i - λ i j , h | ≤ C ⁢ ( γ h ⁢ γ h * ) 1 / α .

3 The Laplace Eigenvalue Problem

Let Ω ⊂ R 2 be a bounded polygonal domain. We are interested in the standard Dirichlet eigenvalue problem for the Poisson equation: find 𝜆 such that, for a nonzero 𝑢, we have

{ - Δ ⁢ u = λ ⁢ u in ⁢ Ω , u = 0 on ⁢ ∂ ⁡ Ω .

We look at a conforming triangulation Ω h and its skeleton ∂ ⁡ Ω h .

We will use two different formulations, namely, the so-called primal and ultraweak formulations (see, for instance, [19, 18, 23]).

3.1 Primal Formulation

The formulation we are considering has been presented in [19] and fits within our setting with the following choices:

U = U 0 × U 1 , U 0 = H 0 1 ⁢ ( Ω ) , U 1 = H - 1 / 2 ⁢ ( ∂ ⁡ Ω h ) , H = H 1 ⁢ ( Ω ) , V = H 1 ⁢ ( Ω h ) , b ⁢ ( u , σ ^ n ; v ) = ( ∇ ⁡ u , ∇ ⁡ v ) Ω h - ( σ ^ n , v ) ∂ ⁡ Ω h , m ⁢ ( u , v ) = ( u , v ) Ω h ,

where as usual the symbol ( ⋅ , ⋅ ) ∂ ⁡ Ω h denotes the action of a functional in H - 1 / 2 ⁢ ( ∂ ⁡ Ω h ) and ( ⋅ , ⋅ ) Ω h the broken L 2 scalar product. We recall the definition of H - 1 / 2 ⁢ ( ∂ ⁡ Ω h ) as

{ τ ∈ ⨂ K H - 1 / 2 ⁢ ( ∂ ⁡ K ) : τ | ∂ ⁡ K = τ ⋅ n | ∂ ⁡ K ⁢ for some ⁢ τ ∈ H ⁢ ( div ; Ω ) ⁢ and all ⁢ K ∈ Ω h } .

We make use of the following discrete spaces for any integer k ≥ 1 :

U h , 0 = S 0 k ⁢ ( Ω h ) , U h , 1 = P k - 1 ⁢ ( ∂ ⁡ Ω h ) ∩ U 1 , U h = U h , 0 × U h , 1 , V h = P k + 1 ⁢ ( Ω h ) .

Here we denote S 0 k ⁢ ( Ω h ) := P k ⁢ ( Ω h ) ∩ C ⁢ ( Ω ¯ ) .

Remark 1

In this section, we are using the standard notation 𝑢 (resp. u h ) for the volumetric part of the solution and σ ^ (resp. σ ^ n ) for the skeleton part. They correspond to u 0 (resp. u 0 , h ) and u 1 (resp. u 1 , h ) in the abstract presentation of the previous section. Analogous notation will be used in the next section for the ultraweak formulation.

The uniform convergence (2.11) is usually proved by employing some a priori estimates of the source problem. The standard estimate for the source problem from [19] reads as follows:

(3.1) ∥ u - u h ∥ H 1 ⁢ ( Ω ) + ∥ σ ^ n - σ ^ h , n ∥ H - 1 / 2 ⁢ ( ∂ ⁡ Ω h ) ≤ C ⁢ inf ( w h , r ^ h , n ) ∈ U h ⁡ ( ∥ u - w h ∥ H 1 ⁢ ( Ω ) + ∥ σ ^ n - r ^ h , n ∥ H - 1 / 2 ⁢ ( ∂ ⁡ Ω h ) ) .

Since we have chosen H = H 1 ⁢ ( Ω ) , the uniform convergence follows from (3.1) as shown in the next proposition.

Proposition 3

Let ( u , σ ^ n ) ∈ U be the solution of source problem (2.1) with right-hand side 𝑓 in H 1 ⁢ ( Ω ) , and assume that 𝑢 belongs to H 1 + s ⁢ ( Ω ) for some s ∈ ( 1 2 , k + 1 ] , where 𝑘 is the order of the approximation introduced above. Then uniform convergence (2.11) holds true.

Proof

Let ( u h , σ ^ h ) ∈ U h be the numerical solution corresponding to our right-hand side 𝑓 in L 2 ⁢ ( Ω ) . Then the regularity assumptions together with the natural error estimates recalled above (see also [19, equation (5.1)] imply

∥ u - u h ∥ H 1 ⁢ ( Ω ) + ∥ σ ^ n - σ ^ h , n ∥ H - 1 / 2 ⁢ ( ∂ ⁡ Ω h ) ≤ C ⁢ h s ⁢ ∥ f ∥ H 1 ⁢ ( Ω ) .

Due to the definitions T F ⁢ ( f ) = u and T F , h ⁢ ( f ) = u h , this implies

∥ ( T F - T F , h ) ⁢ f ∥ H 1 ⁢ ( Ω ) ≤ C ⁢ h s ⁢ ∥ f ∥ H 1 ⁢ ( Ω ) ,

which proves uniform convergence (2.11) with H = H 1 ⁢ ( Ω ) and ρ ⁢ ( h ) ≤ C ⁢ h s . ∎

Corollary 4

Proposition 3 holds also for the following discrete spaces for any odd integer k ≥ 1 :

U h , 0 = S 0 k ⁢ ( Ω h ) , U h , 1 = P k - 1 ⁢ ( ∂ ⁡ Ω h ) ∩ U 1 , V ~ h = P k ⁢ ( Ω h ) .

Proof

The result follows from [5, Theorem 3.5] with the same arguments as in Proposition 3. ∎

We are now in a position to state our main conclusion of this section. For readiness, we consider the case of a simple eigenvalue. Natural modifications apply in case of higher multiplicity.

Theorem 5

Let us consider the DPG primal approximation of the Laplace eigenvalue problem as discussed in Proposition 3. Then the conclusions of Theorems 1 and 2 hold true. In particular, let 𝜆 be a simple eigenvalue of the continuous problem corresponding to an eigenspace 𝐸 belonging to H 1 + s ⁢ ( Ω ) , and let λ h be the approximation of 𝜆 with discrete eigenspace E h . Then we have

(3.2) δ ^ ⁢ ( E , E h ) ≤ C ⁢ h τ , | λ - λ h | ≤ C ⁢ h 2 ⁢ τ

with τ := min ⁡ { s , k } .

Proof

Theorem 1 follows from the convergence in norm (2.11) which we proved in Proposition 3.

In order to verify the rates of convergence shown in (3.2), we have to compute the quantities γ h and γ h * related to the convergence of the DPG primal formulation and of its adjoint formulation, respectively. For the primal formulation, we can estimate γ h by using the optimal bound (3.1), thus obtaining γ h = O ⁢ ( h τ ) , which gives the first in (3.2).

The adjoint problem corresponding to the primal formulation has been considered extensively in [5] for the proof of a duality argument. In our notation, the continuous formulation of the adjoint problem corresponding to (2.6) (see [5, equation (20)]) is: given g ∈ H , find ε * ∈ V and u * = ( u * , σ ^ n * ) ∈ U such that

(3.3) { ( ε * , w ) V + ( ∇ ⁡ u * , ∇ ⁡ w ) Ω h - ( σ ^ n * , w ) ∂ ⁡ Ω h = 0 for all ⁢ w ∈ V , ( ∇ ⁡ ε * , ∇ ⁡ v ) Ω h = ( g , v ) Ω h for all ⁢ v ∈ U 0 , ( τ ^ n , ε * ) ∂ ⁡ Ω h = 0 for all ⁢ τ ^ n ∈ U 1 .

In order to estimate γ h * , we need to consider the discretization of (3.3). It is apparent that the left-hand side of the adjoint problem is the same as the one corresponding the standard primal formulation so that, denoting by ( ε h * , u h * , σ ^ n , h * ) ∈ V h × U h , 0 × U h , 1 the discrete solution, the following quasi-optimal a priori estimate holds true:

(3.4) ∥ ε * - ε h * ∥ V + ∥ u * - u h * ∥ U 0 + ∥ σ ^ n * - σ ^ n , h * ∥ U 1 ≤ inf ( δ , v , τ ^ n ) ∈ V h × U h , 0 × U h , 1 ⁡ ( ∥ ε * - δ ∥ V + ∥ u * - v ∥ U 0 + ∥ σ ^ n * - τ ^ n ∥ U 1 )

It remains to check the regularity of the solution of dual problem (3.3) so that we can get an estimate for γ h * . The estimation of γ h * requires that we consider the restriction of the dual solution operator to the adjoint eigenspace E ′ . That is, we should take g ∈ E ′ in (3.3). We first observe that E ′ = E . This is not surprising since we are dealing with the eigenvalue problem associated with the Laplace operator which is symmetric.

Let 𝚠 be an eigenfunction in 𝐸 associated with the eigenvalue 𝜆. Then it satisfies

{ ( ε , w ) V + ( ∇ ⁡ w , ∇ ⁡ w ) Ω h - ( σ ^ n , w ) ∂ ⁡ Ω h = λ ⁢ ( w , w ) for all ⁢ w ∈ V , ( ∇ ⁡ ε , ∇ ⁡ v ) Ω h = 0 for all ⁢ v ∈ U 0 , ( τ ^ n , ε ) ∂ ⁡ Ω h = 0 for all ⁢ τ ^ n ∈ U 1 ,

where σ ^ n is ∇ ⁡ w ⋅ n on the skeleton ∂ ⁡ Ω h . We will now check that 𝚠 is an eigenfunction of (3.3) as well. Let us take g = w in (3.3). We can define ε * = ( 1 λ ) ⁢ w , and this would satisfy the second equation. Then we can define u * as u * = - 1 + λ λ 2 ⁢ w in Ω and σ ^ n * as σ ^ n * = ∇ ⁡ ( ε * + u * ) ⋅ n on ∂ ⁡ Ω h . By inserting these values in the first equation of (3.3), we get

- 1 λ 2 ⁢ ( ∇ ⁡ w , ∇ ⁡ w ) + 1 λ ⁢ ( w , w ) = 0 for all ⁢ w ∈ V

which is satisfied by 𝚠 since it is the solution of - Δ ⁢ w = λ ⁢ w .

It follows that the regularity of the solution of the dual problem, when restricted to the eigenspace, is exactly the same as for the original problem so that (3.4) gives γ h * = O ⁢ ( h τ ) . Hence, the double order of convergence for the eigenvalues is proved, which concludes our proof. ∎

Remark 2

The comparison of the solution of our original eigenvalue problem and of dual problem (3.3) is consistent with the strong form that was presented in [5, equation (21)] and that can be written as follows in our notation:

- Δ ⁢ ε * = g in ⁢ Ω , ε * = 0 on ⁢ ∂ ⁡ Ω , Δ ⁢ u * = ε * + g in ⁢ Ω , u * = 0 on ⁢ ∂ ⁡ Ω , σ ^ n * = ∇ ⁡ ( ε * + u * ) ⋅ n on ⁢ ∂ ⁡ K ⁢ for all ⁢ K ∈ Ω h .

It must also be noted that, in general, we cannot expect that the dual problem has the same regularity as the original problem. This has been recently shown in [25] where a rigorous analysis has been performed and where a counterexample has been provided. On the other hand, in our case, we can take advantage of the fact that we are solving the dual problem only for g = w , and in this case, the solution of the dual problem coincides, up to a multiplicative constant, with the one of the original problem.

Theorem 5 estimates the eigenfunction error in the energy norm of U 0 . It is interesting to observe that, when the error is estimated in L 2 ⁢ ( Ω ) , then higher order can be achieved. This is stated in the following proposition.

Proposition 6

Under the same assumptions and notation of Theorem 5, the following estimate holds true:

(3.5) δ ^ 0 ⁢ ( E , E h ) ≤ C ⁢ h υ + τ

with υ = min ⁡ { s , 1 } , where δ ^ 0 denotes the gap in the L 2 ⁢ ( Ω ) norm.

Proof

The result is a consequence of the analogous one valid for the source problem which has been proved in [5, Theorem 3.1] in the case of a convex domain. This is a standard Aubin–Nitche duality argument. The general case follows by inspecting the proof of [5, Theorem 3.1] dealing with [5, Assumption 2.9] where the regularity of the dual (adjoint) problem is discussed. Reference [5] studies the case of full regularity s = 1 ; when s < 1 , exactly the same arguments give the result with C 3 ⁢ ( h ) = h s . ∎

We now switch back momentarily to the notation of the abstract setting, where the component 𝑢 of the solution was denoted by u 0 and the component σ ^ by u 1 . Then we observe that the convergence estimates in the first of (3.2) and in (3.5) imply that, given u 0 ∈ E , there exists u 0 , h ∈ E h such that

(3.6) ∥ u 0 - u 0 , h ∥ U 0 ≤ C ⁢ h τ ⁢ ∥ u 0 ∥ H 1 + s ⁢ ( Ω ) , ∥ u 0 - u 0 , h ∥ L 2 ⁢ ( Ω ) ≤ C ⁢ h τ + υ ⁢ ∥ u 0 ∥ H 1 + s ⁢ ( Ω ) .

When also the other component of the eigenfunction is considered, we have the following a priori estimate:

(3.7) ∥ u - u h ∥ U ≤ C ⁢ h τ ⁢ ∥ u 0 ∥ H 1 + s ⁢ ( Ω ) ,

where the components u 1 and u 1 , h of 𝑢 and u h are the ones corresponding to u 0 and u 0 , h in (2.7) and (2.9), respectively.

3.2 Ultraweak Formulation

The DPG ultraweak formulation for the Laplace eigenvalue problem fits within our abstract setting with the following choices (see [18, 23] for more details):

U = U 0 × U 1 , U 0 = L 2 ⁢ ( Ω ) , U 1 = L 2 ⁢ ( Ω ) 2 × H 0 1 / 2 ⁢ ( ∂ ⁡ Ω h ) × H - 1 / 2 ⁢ ( ∂ ⁡ Ω h ) , H = U 0 , V = H 1 ⁢ ( Ω h ) × H ⁢ ( div ; Ω h ) ,

b ⁢ ( u , σ , u ^ , σ ^ n ; v , τ ) = ( σ , τ ) Ω h - ( u , div ⁢ τ ) Ω h + ( u ^ , τ ⋅ n ) ∂ ⁡ Ω h - ( σ , ∇ ⁡ v ) Ω h + ( v , σ ^ n ) ∂ ⁡ Ω h , m ⁢ ( u , σ , u ^ , σ ^ n ; v , τ ) = ( u , v ) Ω h ,

where the space H 0 1 / 2 ⁢ ( ∂ ⁡ Ω h ) is defined as

{ w ^ ∈ ⨂ K H 1 / 2 ⁢ ( ∂ ⁡ K ) : w ^ | ∂ ⁡ K = w | ∂ ⁡ K ⁢ for some ⁢ w ∈ H 0 1 ⁢ ( Ω ) ⁢ and all ⁢ K ∈ Ω h }

and H 1 ⁢ ( Ω h ) and H ⁢ ( div ; Ω h ) denote broken functional spaces on the mesh Ω h .

We choose the following discrete spaces with k ≥ 0 :

(3.8) U h := P k ⁢ ( Ω h ) × P k ⁢ ( Ω h ; R 2 ) × S 0 k + 1 ⁢ ( ∂ ⁡ Ω h ) × P k ⁢ ( ∂ ⁡ Ω h ) , V h := P k + 2 ⁢ ( Ω h ) × P k + 2 ⁢ ( Ω h ; R 2 ) ,

where S 0 k + 1 ⁢ ( ∂ ⁡ Ω h ) is defined as follows:

S 0 k + 1 ⁢ ( ∂ ⁡ Ω h ) := γ 0 ⁢ ( S 0 k + 1 ⁢ ( Ω h ) ∩ H 0 1 ⁢ ( Ω ) ) .

Here γ 0 denotes the canonical trace operator from H 1 ⁢ ( Ω ) to H 1 / 2 ⁢ ( ∂ ⁡ Ω h ) .

Also in this case, we can use the a priori estimates in order to show uniform convergence (2.11).

Proposition 7

Let T F : H → H be the solution operator associated with the continuous problem and T F , h its discrete counterpart as defined in (2.8) and (2.10), respectively. Assume that the solution 𝑢 of the Poisson problem with 𝑓 in L 2 ⁢ ( Ω ) belongs to H 1 + s ⁢ ( Ω ) for some s ∈ ( 1 2 , k + 1 ] , where 𝑘 is the order of approximation used in (3.8). Then the convergence in norm (2.11) holds true.

Proof

The a priori error analysis presented, for instance, in [21, Corollary 6] reads ∥ u - u h ∥ U 0 ≤ C ⁢ h s ⁢ ∥ f ∥ L 2 ⁢ ( Ω ) , which implies uniform convergence (2.11). ∎

The rate of convergence that follows naturally from the a priori error estimate for the source problem is presented in the following theorem.

Theorem 8

Let us consider the DPG ultraweak approximation of the Laplace eigenvalue problem as discussed in Proposition 7. Then the conclusions of Theorems 1 and 2 hold true. In particular, let 𝜆 be a simple eigenvalue of the continuous problem corresponding to an eigenspace 𝐸 belonging to H 1 + s ⁢ ( Ω ) , and let λ h be the approximation of 𝜆 with discrete eigenspace E h . Then we have δ ^ ⁢ ( E , E h ) ≤ C ⁢ h τ and | λ - λ h | ≤ C ⁢ h 2 ⁢ τ with τ = min ⁡ { s , k + 1 } .

Proof

We omit the technical detail of the proof that follows the same lines as the proof of Theorem 5. In particular, the estimates are obtained by inspecting the a priori estimates of the ultraweak formulation and of its adjoint under our regularity assumptions. The a priori estimates of the ultraweak formulation have been proved in [18] (see discussion before Corollary 4.1) and read as follows:

∥ u - u h ∥ L 2 ⁢ ( Ω ) + ∥ σ - σ h ∥ L 2 ⁢ ( Ω ) + ∥ u ^ - u ^ h ∥ H 0 1 / 2 ⁢ ( ∂ ⁡ Ω h ) + ∥ σ ^ n - σ ^ n , h ∥ H - 1 / 2 ⁢ ( ∂ ⁡ Ω h ) ≤ C ⁢ h τ ⁢ ( ∥ u ∥ H 1 + τ ⁢ ( Ω ) + ∥ σ ∥ H τ ⁢ ( Ω ) + ∥ div ⁢ σ ∥ H τ ⁢ ( Ω ) ) . ∎

It is also possible to introduce a slightly different lowest order approximation for the ultraweak formulation. The discretization reads as follows:

U ~ h = P 0 ⁢ ( Ω h ) × P 0 ⁢ ( Ω h ; R 2 ) × S 0 1 ⁢ ( ∂ ⁡ Ω h ) × P 0 ⁢ ( ∂ ⁡ Ω h ) , V ~ h = P 1 ⁢ ( Ω h ) × RT 0 P ⁢ W ⁢ ( Ω h ) ,

where RT 0 P ⁢ W ⁢ ( Ω h ) denotes the discontinuous Raviart–Thomas space of lowest degree.

Corollary 9

With the same assumptions as in Theorem 8, it holds for the lowest order case that

δ ^ ⁢ ( E , E h ) ≤ C ⁢ h τ , | λ - λ h | ≤ C ⁢ h 2 ⁢ τ .

Proof

As in the case of Theorems 5 and 8, the result is obtained by inspecting the a priori estimates of the ultraweak formulation and of its adjoint under our regularity assumptions. The a priori estimates for this choice of discrete spaces can be found in [13, Theorem 3.3]. ∎

For reasons that will become clearer in the next section, it will be useful to have a higher order estimate of the U 0 component of the solution in the spirit of what we obtained in (3.6) for the primal formulation. This property can be achieved by augmenting the approximating space U 0 , h along the lines of what was proposed in [21].

We then choose the following discrete spaces with k ≥ 0 :

U h := P k + 1 ⁢ ( Ω h ) × P k ⁢ ( Ω h ; R 2 ) × S 0 k + 1 ⁢ ( ∂ ⁡ Ω h ) × P k ⁢ ( ∂ ⁡ Ω h ) , V h := P k + 2 ⁢ ( Ω h ) × P k + 2 ⁢ ( Ω h ; R 2 ) ,

where the order of the polynomials approximating U 0 is raised from 𝑘 to k + 1 .

We now revert back to the notation of the abstract setting where the symbols 𝑢 and u h refer to pairs ( u 0 , u 1 ) and ( u 0 , h , u 1 , h ) in 𝑈 and U h , respectively.

We use the improved a priori estimates obtained in [21, Theorem 10], which reads

∥ u 0 - u 0 , h ∥ U 0 ≤ C ⁢ h s + s ′ ⁢ ∥ f ∥ L 2 ⁢ ( Ω ) ,

where s ′ ∈ ( 1 2 , 1 ] denotes the regularity shift of an auxiliary problem used for a duality argument. This implies now the following result.

Theorem 10

With the same assumptions as in Theorem 8, the augmented formulation provides the following a priori error estimates. Given u 0 ∈ E and its corresponding u = ( u 0 , u 1 ) ∈ U , there exists u 0 , h ∈ E h and its corresponding u h = ( u 0 , h , u 1 , h ) ∈ U h such that

∥ u - u h ∥ U ≤ C ⁢ h τ ⁢ ∥ u 0 ∥ H 1 + s ⁢ ( Ω ) , ∥ u 0 - u 0 , h ∥ U 0 ≤ C ⁢ h τ + s ′ ⁢ ∥ u 0 ∥ H 1 + s ⁢ ( Ω )

with τ = min ⁡ { s , k + 1 } and s ′ ∈ ( 1 2 , 1 ] is the regularity shift defined in [21, equation (14)].

4 A Posteriori Error Analysis

In this section, we present and discuss two error estimators that can be used in the framework of a posteriori analysis and adaptive schemes.

4.1 The Natural Error Estimator

We start with the most natural error estimator, associated with the energy residual, that has been considered in [10] for the source problem. The residual is the component ε h of the solution to problem (2.6). In the setting of [10], we can consider two operators B , M : U → V ′ defined as

( B ⁢ u ) ⁢ ( v ) := b ⁢ ( u , v ) for all ⁢ u ∈ U ⁢ and all ⁢ v ∈ V , ( M ⁢ u ) ⁢ ( v ) := m ⁢ ( u 0 , v ) for all ⁢ u ∈ U ⁢ and all ⁢ v ∈ V ,

where we are adopting the splitting u = ( u 0 , u 1 ) ∈ U = U 0 × U 1 as considered in Section 2. We denote the operator norm as C B and C M . The natural error indicator studied theoretically in [10] is the global indicator

(4.1) η = ∥ ε h ∥ = ∥ λ h ⁢ M ⁢ u h - B ⁢ u h ∥ V ′

(see (2.6)), where the definition of M ⁢ u h makes use of the component u 0 , h of u h ; the practical implementation of an adaptive scheme based on 𝜂 employs a localized version of it.

With natural modifications of the analysis of [10], global efficiency and reliability can be proved. Both properties rely on the following (usual) higher order term λ ⁢ u 0 - λ h ⁢ u 0 , h (see Section 4.3).

Proposition 11

Proposition 11 (Reliability and Efficiency)

Let us examine an eigenvalue 𝜆 of problem (2.7) of multiplicity one with eigenfunction u ∈ U and the corresponding discrete eigenpair ( λ h , u h ) . Assume that (2.2), (2.3) and (2.5) are satisfied. Then the following reliability and efficiency estimates hold true:

C 1 ⁢ ∥ u - u h ∥ U ≤ η 2 + ( ∥ λ ⁢ ( M ⁢ u ) ∘ ( 1 - Π ) ∥ V * + η ⁢ ∥ Π ∥ ) 2 + C M ⁢ ∥ λ ⁢ u 0 - λ h ⁢ u 0 , h ∥ , η 2 ≤ C B 2 ⁢ ∥ u - u h ∥ U 2 + C M 2 ⁢ ∥ λ ⁢ u 0 - λ h ⁢ u 0 , h ∥ 2 .

Proof

We define the error e := u - u h , the error representation function 𝜀 by

(4.2) ( ε , y ) V = λ h ⁢ m ⁢ ( u 0 , h , v ) - b ⁢ ( u h , v ) for all ⁢ v ∈ V

and its approximation ε h by

(4.3) ( ε h , y h ) V = λ h ⁢ m ⁢ ( u 0 , h , v h ) - b ⁢ ( u h , v h ) for all ⁢ v h ∈ V h .

From inf-sup condition (2.3), it follows

C 1 ⁢ ∥ e ∥ U ≤ ∥ B ⁢ e ∥ V * ≤ ∥ ε ∥ V + ∥ M ⁢ ( λ ⁢ u - λ h ⁢ u h ) ∥ V ′ ≤ ∥ ε ∥ V + C M ⁢ ∥ λ ⁢ u 0 - λ h ⁢ u 0 , h ∥ .

From (4.2) and (4.3), we see that δ := ε - ε h ⟂ V h so that the Pythagoras theorem gives

(4.4) ∥ ε ∥ V 2 = ∥ ε h ∥ V 2 + ∥ δ ∥ V 2 .

Noting that Π ⁢ δ ∈ V h ⟂ δ and from (2.5), we can conclude that

∥ δ ∥ V 2 = ( δ , δ - Π ⁢ δ ) V = ( ε - ε h , δ - Π ⁢ δ ) V = ( ε , δ - Π ⁢ δ ) V + ( ε h , Π ⁢ δ ) V .

The properties of Π lead to

( ε , δ - Π ⁢ δ ) V = b ⁢ ( u - u h , δ - Π ⁢ δ ) = λ ⁢ ( M ⁢ u ) ⁢ ( δ - Π ⁢ δ ) .

From the previous estimate, it follows

∥ δ ∥ V 2 = λ ⁢ ( M ⁢ u ) ⁢ ( δ - Π ⁢ δ ) + ( ε h , Π ⁢ δ ) V ≤ ( ∥ λ ⁢ ( M ⁢ u ) ∘ ( 1 - Π ) ∥ V ′ + ∥ ε h ∥ V ⁢ ∥ Π ∥ ) ⁢ ∥ δ ∥ V .

Finally,

C 1 ⁢ ∥ e ∥ U ≤ ∥ ε h ∥ V 2 + ( ∥ λ ⁢ ( M ⁢ u ) ∘ ( 1 - Π ) ∥ V ′ + ∥ ε h ∥ V ⁢ ∥ Π ∥ ) 2 + C M ⁢ ∥ λ ⁢ u 0 - λ h ⁢ u 0 , h ∥ .

For the efficiency, we can use decomposition (4.4) and (2.3) so that we obtain easily

∥ ε h ∥ V 2 = ∥ ε ∥ V 2 - ∥ δ ∥ V 2 ≤ ∥ B ⁢ e ∥ V ′ 2 + C M 2 ⁢ ∥ λ ⁢ u 0 - λ h ⁢ u 0 , h ∥ 2 ≤ C B 2 ⁢ ∥ e ∥ U 2 + C M 2 ⁢ ∥ λ ⁢ u 0 - λ h ⁢ u 0 , h ∥ 2 . ∎

Corollary 12

Proposition 11 holds true for the primal and ultraweak formulation of the Laplace eigenvalue problem that we have discussed in the previous section.

Proof

The hypotheses stated in Proposition 11 are classical in the setting of the DPG approximation of the Laplace problem; see, for instance, [13, Section 3.2] and [10, Section 3.1]. ∎

Remark 3

The occurrence of a nonlinear term like λ ⁢ u 0 - λ h ⁢ u 0 , h is typical when translating a posteriori analysis from the source problem to the eigenvalue problem. Although this term is not suited for the standard AFEM setting, it is generally a higher order term. We will comment more on this fact in Section 4.3.

4.2 An Alternative Error Estimator

In the case of lowest order approximations, we now present an error estimator which depends only on the jump terms and that will turn out to be equivalent to the natural error estimator 𝜂. Therefore, for this, we use arguments similar to the ones in [24] for the source problem. The proof relies on special properties of the Crouzeix–Raviart spaces, which are defined as follows:

CR 1 ⁢ ( Ω h ) := { v ∈ P 1 ⁢ ( Ω h ) ∣ v ⁢ is continuous at ⁢ mid ⁡ ( E ) ⁢ for all ⁢ E ∈ E ⁢ ( Ω h ) } , CR 0 1 ⁢ ( Ω h ) := { v ∈ CR 1 ⁢ ( Ω h ) ∣ v ⁢ ( mid ⁡ ( E ) ) = 0 ⁢ for all ⁢ E ∈ E ⁢ ( ∂ ⁡ Ω h ) } ,

where E ⁢ ( Ω h ) and E ⁢ ( ∂ ⁡ Ω h ) denote the sets of interior and boundary edges of the triangulation, respectively.

This lemma from [24, Lemma 3.2] shows a general orthogonality relationship between the Crouzeix–Raviart spaces and continuous P 1 spaces and is crucial for the equivalent statements.

Lemma 13

Any w CR ∈ CR 0 1 ⁢ ( Ω h ) with the L 2 orthogonality ∇ h ⁡ w CR ⟂ ∇ ⁡ S 0 1 ⁢ ( Ω h ) satisfies

| | | w CR | | | pw 2 ≈ ∑ T ∈ Ω h | T | 1 / 2 ⁢ ∑ E ∈ E ⁢ ( T ) ∥ [ ∇ h ⁡ w CR ] E ⋅ ν E ∥ L 2 ⁢ ( E ) 2 ≈ ∑ T ∈ Ω h | T | 1 / 2 ⁢ ∑ E ∈ E ⁢ ( T ) ∥ [ ∇ h ⁡ w CR ] E ⋅ τ E ∥ L 2 ⁢ ( E ) 2 ,

where ∇ h denotes the broken gradient and | | | ⋅ | | | pw := ∥ ∇ h ⋅ ∥ L 2 ⁢ ( Ω ) denotes the broken energy norm. In addition, [ v ] E := v T + - v T - defines the jump of v ∈ H 1 ⁢ ( Ω h ) on any inner side E ∈ E ⁢ ( Ω h ) .

We recall the lowest order approximation of the primal formulation, where we use the standard notation u h for u 0 , h and σ ^ h for u 1 , h : find λ h ∈ C such that, for ( u h , σ ^ h ) ∈ U h and ε h ∈ V h , it holds

(4.5) ( ε h , v h ) V + ( ∇ ⁡ u h , ∇ ⁡ v h ) Ω h - ⟨ σ ^ h , v h ⟩ ∂ ⁡ Ω h = λ h ⁢ ( u h , v h ) Ω h for all ⁢ v h ∈ V h , ( ∇ ⁡ z h , ∇ ⁡ ε h ) Ω h - ⟨ t ^ h , ε h ⟩ ∂ ⁡ Ω h = 0 for all ⁢ ( z h , t ^ h ) ∈ U h ,

where U h and V h are defined as follows:

U h , 0 = S 0 1 ⁢ ( Ω h ) , U h , 1 = P 0 ⁢ ( ∂ ⁡ Ω h ) ∩ U 1 , U h = U h , 0 × U h , 1 , V h = P 1 ⁢ ( Ω h ) .

The estimator we are looking at has been introduced in [24] for the source problem. We define the alternative error estimator η ¯ as follows:

(4.6) η ¯ 2 = ∑ T ∈ Ω h | T | 1 / 2 ⁢ ∑ E ∈ E ⁢ ( T ) ∥ [ ∇ h ⁡ ε h ] E ∥ L 2 ⁢ ( E ) 2 .

The equivalence between the alternative error estimator and the one discussed in the previous section is stated in the following theorem and uses orthogonality arguments, which only hold in the lowest order case.

Theorem 14

Let ( u h , σ h ) ∈ U h and ε h ∈ V h be the solution of discrete primal problem (4.5). Then the two error estimators defined in (4.1) and (4.6) are equivalent.

Proof

We follow here the arguments of [24, Theorem 4.1].

For any s ^ 0 ∈ U h , 1 , we have 0 = - b ⁢ ( 0 , s ^ 0 ; ε h ) = ( s ^ 0 , ε h ) ∂ ⁡ Ω h . It follows that ∫ E [ ε h ] E = 0 for all E ∈ ∂ ⁡ Ω h so that ε h belongs to CR 0 1 ⁢ ( Ω h ) . If we choose ( w , 0 ) ∈ U h as test function, then ( ∇ ⁡ w C , ∇ ⁡ ε h ) = 0 for all w C ∈ S 0 1 ⁢ ( Ω h ) and the second property follows.

So ε h satisfies the assumptions of Lemma 13, and we can conclude from (4.6),

| | | ε h | | | pw 2 ≈ ∑ T ∈ Ω h ∥ T ∥ 1 / 2 ⁢ ∑ E ∈ E ⁢ ( T ) ∥ [ ∇ h ⁡ ε h ] E ∥ L 2 ⁢ ( E ) 2 = η ¯ 2 .

Now, from the discrete Friedrichs inequality for Crouzeix–Raviart spaces [7], it follows

η ¯ 2 ≈ | | | ε h | | | pw 2 ≤ ∥ ε h ∥ L 2 ⁢ ( Ω h ) 2 + | | | ε h | | | pw 2 = η 2 ≲ | | | ε h | | | pw 2 ≈ η ¯ 2 . ∎

We observe, in particular, that the alternative error estimator η ¯ inherits the global efficiency and reliability from 𝜂.

To conclude this section, we remark that the alternative error estimator can be defined for the ultraweak formulation as well. The formulation we are considering is: find λ h ∈ C and ( u h , σ h , σ ^ n , h , u ^ h ) ∈ U h with u h ≠ 0 such that, for ε h = ( v h , τ h ) ∈ V h , it holds

(4.7) { ( ε h , ( v , τ ) ) V + b ⁢ ( u h , σ h , σ ^ n , h , u ^ h ; v , τ ) = λ h ⁢ m ⁢ ( u h , v ) for all ⁢ ( v , τ ) ∈ V h , b ⁢ ( w , r , s , t ; ε h ) = 0 for all ⁢ ( w , r , s , t ) ∈ U h ,

where the discrete spaces are given by

U h = P 0 ⁢ ( Ω h ) × P 0 ⁢ ( Ω h ; R 2 ) × S 0 1 ⁢ ( ∂ ⁡ Ω h ) × P 0 ⁢ ( ∂ ⁡ Ω h ) and V h = P 1 ⁢ ( Ω h ) × RT 0 ⁢ ( Ω h ) .

We can now define the alternative error estimator for the ultraweak formulation analogously as we have done for the primal formulation as follows:

(4.8) η ~ 2 := ∑ T ∈ Ω h | T | 1 / 2 ⁢ ∑ E ∈ E ⁢ ( T ) ∥ [ ∇ h ⁡ v h ] E ∥ L 2 ⁢ ( E ) 2 = ∑ T ∈ Ω h | T | 1 / 2 ⁢ ∑ E ∈ E ⁢ ( T ) ∥ [ τ h ] E ∥ L 2 ⁢ ( E ) 2 .

Also in this case, the alternative error estimator is equivalent to the natural one with arguments similar to [24, Theorem 4.4].

Theorem 15

Let ε h = ( v h , τ h ) ∈ V h and ( u h , σ h , σ ^ n , h , u ^ h ) ∈ U h be the solution of discrete ultraweak problem (4.7). Then the two error estimators for the ultraweak formulation defined in (4.1) and (4.8) are equivalent.

Proof

Similarly to the primal case, we first check that the assumptions of Lemma 13 are fulfilled.

We can choose as test functions ( w ~ , r ~ , 0 , s ~ ) ∈ U h . It follows from (4.7) that v h belongs to CR 0 1 ⁢ ( Ω h ) , that div ⁡ τ h = 0 and that τ h + ∇ h ⁡ v = 0 . Now we choose as test function ( 0 , 0 , s ~ , 0 ) with s ~ ∈ S 0 1 ⁢ ( Ω h ) . After integration by parts, it follows

( τ ⋅ n , s ~ ) ∂ ⁡ Ω h = ( div h ⁡ τ , s ~ ) L 2 ⁢ ( Ω ) + ( τ , ∇ h ⁡ s ~ ) L 2 ⁢ ( Ω ) = - ( ∇ h ⁡ v , ∇ h ⁡ s ~ ) L 2 ⁢ ( Ω ) = 0 ,

where div h denotes the broken divergence operator.

From the identity τ h = - ∇ h ⁡ v h just shown, it follows

∥ v h ∥ H 1 ⁢ ( Ω h ) 2 ≤ ∥ τ h ∥ H ⁢ ( div , Ω h ) 2 + ∥ v h ∥ H 1 ⁢ ( Ω h ) 2 = ∥ ( τ h , v h ) ∥ V 2 = ∥ τ h ∥ L 2 ⁢ ( Ω ) 2 + ∥ div ⁡ τ h ∥ L 2 ⁢ ( Ω ) 2 + ∥ v h ∥ H 1 ⁢ ( Ω h ) 2 = ∥ ∇ ⁡ v h ∥ L 2 ⁢ ( Ω ) 2 + ∥ v h ∥ H 1 ⁢ ( Ω h ) 2 ≤ 2 ⁢ ∥ v h ∥ H 1 ⁢ ( Ω h ) 2 .

Now we can use Lemma 13 together with the discrete Friedrichs inequality to get the result

η ~ 2 ≈ | | | v h | | | 2 ≤ ∥ v h ∥ H 1 ⁢ ( Ω h ) 2 ≤ ∥ ( τ h , v h ) ∥ V 2 = η 2 ≤ 2 ⁢ ∥ v h ∥ H 1 ⁢ ( Ω h ) 2 ≲ | | | v h | | | 2 ≈ η ~ 2 . ∎

Hence, also in this case, η ~ inherits the global efficiency and reliability from 𝜂.

4.3 The Nonlinear Term ∥ λ ⁢ u 0 - λ h ⁢ u 0 , h ∥

We show in this section how to compare the nonlinear term ∥ λ ⁢ u 0 - λ h ⁢ u 0 , h ∥ to the error ∥ u - u h ∥ U . We are going to use the following identity:

(4.9) λ ⁢ u 0 - λ h ⁢ u 0 , h = λ ⁢ ( u 0 - u 0 , h ) + u 0 , h ⁢ ( λ - λ h ) .

A standard way to show that this is a higher order term is to observe that ∥ u 0 - u 0 , h ∥ is converging in the L 2 ⁢ ( Ω ) norm faster than the error in the energy norm and that | λ - λ h | is converging with double order.

In the DPG approximations that we are discussing, we can see that this property is valid for | λ - λ h | , while particular attention has to be paid for the term ∥ u 0 - u 0 , h ∥ . Indeed, the a priori estimates for the primal formulation recalled in (3.7) and (3.6) guarantee that ∥ u 0 - u 0 , h ∥ L 2 ⁢ ( Ω ) is of higher order with respect to ∥ u - u h ∥ U , while this is not always the case for the ultraweak formulation. When the augmented formulation is used, Theorem 10 guarantees that we have the desired property. We summarize the obtained results in the following statement.

Proposition 16

In the case of the primal DPG formulation presented in Section 3.1 and of the augmented ultraweak DPG formulation presented in Section 3.2, the nonlinear term ∥ λ ⁢ u 0 - λ h ⁢ u 0 , h ∥ L 2 ⁢ ( Ω ) is of order O ⁢ ( h τ + υ ) when ℎ goes to zero, with respect to the energy norm error ∥ u - u h ∥ U which is of order O ⁢ ( h τ ) , where 𝜏 and 𝜐 are defined in Theorems 5 and 8, in Proposition 6 and in Theorem 10.

Proof

The result follows from identity (4.9) and the a priori estimates presented in (3.7), (3.6) and in Theorem 10. ∎

5 Numerical Results

This section presents selected numerical experiments in two dimensions for the primal and ultraweak formulations. We implemented the lowest order method based on [14, 24]. For the adaptive mesh refinement, we used the classical AFEM algorithm [12] with Dörfler marking and bulk parameter θ = 0.5 . For the higher order formulations, we used NETGEN/NGSOLVE [28, 29]. In case of the ultraweak formulation, we used the discretization from (3.8). In Figure 3, we show some of our mesh obtained after four refinements. As expected, the meshes are strongly refined where singular solutions are expected. The meshes are generated for the actual eigenvalue problem with respect to the estimator of the first eigenpair. We always solved for the first 20 eigenpairs.

Figure 3

Adaptive meshes after four iterations

(a)

Slit domain adaptive mesh for primal DPG

(b)

L-shaped domain adaptive mesh for primal DPG

5.1 Numerical Results on the Square Domain

Our first domain is the convex square domain, which is defined as Ω = [ 0 , 1 ] 2 . For this domain, the exact solution is well known. Here we compare our solution with the first eigenvalue λ 1 = 2 ⁢ π 2 . In Figure 4, we can clearly see that the methods show the expected convergence rates.

Figure 4

Convergence rates on the square domain, uniform refinement (p = primal, u = ultraweak)

5.2 Numerical Results on the L-Shaped Domain

On the non-convex L-shaped domain Ω = ( - 1 , 1 ) 2 ∖ ( [ 0 , 1 ) × ( - 1 , 0 ] ) , we used the reference values

λ 1 = 9.639723844871536 and λ 5 = 31.91263 .

These two eigenvalues correspond to singular eigenspaces that belong to H 1 + s ⁢ ( Ω ) with s < 1 2 . For this reason, we expect a convergence rate of 2 3 in terms of the number of degrees of freedom when using a uniform mesh refinement. This rate of convergence is clearly detected for λ 1 in Figure 5, while in the case of λ 5 a pre-asymptotic convergence is detected: the rate of convergence is approximately 0.78 in the last iteration of Figure 6, but it can be seen that the rate is degenerating as the mesh is refined. For bulk parameter θ = 0.5 , all adaptive methods show optimal convergence rates.

For the higher order methods, Figure 7 shows a similar behavior as in the lowest order case. Moreover, the convergence rate can be restored in the adaptive case (Figure 8).

Figure 5

Convergence rates for the L-shaped domain, first eigenvalue

Figure 6

Convergence rates for the L-shaped domain, fifth eigenvalue

Figure 7

Convergence rates for higher order on L-shaped domain (p = primal, u = ultraweak)

Figure 8

Adaptive convergence rates for the L-shaped domain for higher order (p = primal, u = ultraweak)

5.3 Numerical Results on the Slit Domain

On the non-convex slit domain Ω = ( - 1 , 1 ) 2 ∖ ( [ 0 , 1 ) × { 0 } ) , we used λ 1 = 8.371329711 as reference solution. For the first eigenvalue, we expect a convergence rate of 1 2 when the mesh is refined uniformly. Like for the L-shape domain, the adaptive methods can recover the optimal convergence rate.

Figure 9

Convergence rates for the slit domain

5.4 Higher Order Term

In our next numerical simulation, we check in the lowest order case the statement of Proposition 16 about the higher order term appearing in our a posteriori estimates. In order to calculate the higher order term, we computed a reference solution on a fine mesh with about a million of degrees of freedom. In Figure 10, we can appreciate that the term ∥ λ ⁢ u 0 - λ h ⁢ u 0 , h ∥ is actually of higher order and converges twice as fast as the eigenfunction in both cases that we have considered. The rate is actually stabilizing about the value of 1, even if, on the slit domain, the convergence is faster in the pre-asymptotic regime.

Figure 10

Convergence higher order term

5.5 Efficiency Ratio

Our last numerical test concerns the efficiency ratio, defined as η ∥ u ~ - u ∥ U . We used the same reference solution that we computed for the higher order term calculation. In Table 1, we can see that, for the L-shaped domain, we have an efficiency ratio between 9 and 10. For the slit domain, we have a ratio between 14 and 19.

Table 1

Efficiency ratio for primal error estimator 𝜂

DoF	Efficiency ratio (L-shaped)	DoF	Efficiency ratio (slit)
120	7.759	160	8.406
290	7.557	375	8.218
535	7.718	755	8.852
1145	8.914	1365	10.191
2005	9.064	2390	11.435
3875	9.290	4330	12.757
6765	9.714	7490	14.161
12565	9.843	13390	16.884
23075	9.471	23020	17.713
43055	10.200	41345	18.275
79080	9.998	72360	17.590
140000	9.258	124155	14.341

Funding source: Deutsche Forschungsgemeinschaft

Award Identifier / Grant number: BE 6511/1-1

Funding source: Gruppo Nazionale per il Calcolo Scientifico

Award Identifier / Grant number: IMATI/CNR

Award Identifier / Grant number: PRIN/MIUR

Funding statement: The first author gratefully acknowledge support by the DFG in the Priority Program SPP 1748 Reliable simulation techniques in solid mechanics, Development of non-standard discretization methods, mechanical and mathematical analysis under the project number BE 6511/1-1. The second author is member of the INdAM Research group GNCS and his research is partially supported by IMATI/CNR and by PRIN/MIUR.

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Received: 2022-03-18

Accepted: 2022-03-21

Published Online: 2022-05-26

Published in Print: 2023-01-01

This work is licensed under the Creative Commons Attribution 4.0 International License.

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https://doi.org/10.1515/cmam-2022-0069

Keywords for this article

Eigenvalue Problems; DPG Methods

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