DeepAI

# Quasi-optimal adaptive mixed finite element methods for controlling natural norm errors

For a generalized Hodge--Laplace equation, we prove the quasi-optimal convergence rate of an adaptive mixed finite element method controlling the error in the natural mixed variational norm. In particular, we obtain new quasi-optimal adaptive mixed methods for the scalar Poisson, vector Poisson, and Stokes equations. Comparing to existing adaptive mixed methods, the new methods control errors in both two variables.

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## 1. Introduction

Adaptive finite element method(AFEM) has been an active research area since the pioneering work [5], see, e.g., [45, 7, 37] for an thorough introduction. Comparing to finite element methods using quasi-uniform meshes, AFEMs can achieve quasi-optimal convergence rate by producing a sequence of graded meshes resolving singularity arising from irregular data of differential equations and domains with corners or slits. Typically, AFEM can be described by the feedback loop

 SOLVE⟶ESTIMATE⟶MARK⟶REFINE.

Given a conforming mesh Solve returns the finite element solution of the discrete problem on . Estimate returns a collection of error indicators . MARK selects a subset of using the information from . A conforming subtriangulation is then obtained by applying REFINE to and SOLVE is called on . Despite the popularity of AFEMs in practice, [6] for the one-dimensional boundary value problem had been the only convergence result of AFEMs for a long time. Using a bulk chasing marking strategy in MARK, Dörfler [23] first proved that the Lagrange element solution converges to the exact solution in the energy norm for Poisson’s equation in provided the initial mesh is fine enough. Readers are referred to [33, 9, 42, 17] and references therein for further important progress in the analysis of convergence and optimality of AFEMs for symmetric and positive-definite elliptic problems. Of particular relevance in this paper is [25], where the authors used weak convergence technique to prove the quasi-optimal convergence rate of AFEMs for nonsymmetric and nonlinear elliptic problems.

The mixed finite element method(MFEM) is designed to numerically solve systems of partial differential equations arising from elasticity, fluids, electromagnetism, computational geometry etc. In contrast to AFEMs based on positive-definite formulations, the difficulty in convergence and optimality analysis of adaptive mixed finite element methods (AMFEMs) are two-fold. First, the a posteriori error analysis hinges on delicate decomposition results and possibly bounded commuting quasi-interpolations onto a sequence of finite elements spaces, see, e.g.,

[1, 39, 22]. In addition, those quasi-interpolations are even required to locally preserve finite element functions when deriving discrete reliability, see, e.g., [21, 46]. Second, the exact solution of a system of equations is generally only a critical point of some variational principle. Hence is not orthogonal to and a technical quasi-orthogonality is indispensable, see, e.g., [18, 8].

Consider the popular model problem for the analysis of MFEMs: Find such that

 (1.1) ⟨σ,τ⟩−⟨divτ,u⟩ =0,τ∈H(div;Ω), ⟨divσ,v⟩ =⟨f,v⟩,v∈L2(Ω).

In fact (1.1) is the mixed formulation of Poisson’s equation. Let be the finite element solutions and meshes produced by some AMFEM for (1.1) using Raviart–Thomas(RT) or Brezzi–Douglas–Marini(BDM) elements, see [38, 11]. Under mild assumptions, it has been shown in e.g., [18, 8, 29, 27] that converges to in the -norm with quasi-optimal convergence rate. As far as we know, existing AMFEMs for Poisson’s equation are not able to control the error because most error indicators and quasi-orthogonality in literature are not designed for the natural -norm. This limitation seems not so severe for Poisson’s equation, since is trivially controlled by and the scalar variable is practically less important than the flux . However, there are still several works on the a posteriori -error estimates of mixed methods for (1.1), see, e.g., [10, 14].

Poisson’s equation is a special case of the Hodge Laplace equation , which is the model problem in the theory of fintie element exterior calculus (FEEC) developed by Arnold, Falk, and Winther [3, 4]. Here is the exterior derivative for differential forms and is the adjoint operator of . In general, the Hodge Laplace equation is solved by the mixed method (2.4) in FEEC literature. Adaptivity in FEEC has been an active research area in recent years. Using their regular decomposition and commuting quasi-interpolation, Demlow and Hirani [22] developed the first reliable a posteriori error estimator for controlling the error of the mixed method (2.4). At the same time, Falk and Winther [24] constructed a technical local bounded commuting interpolation connecting the de Rham complex (2.10) and its finite element subcomplex. Using these ingredients, [21, 19, 30, 27] recently developed quasi-optimal AMFEMs for problems posed on the de Rham complex. For the Hodge Laplace equation, we [30] developed an AMFEM for controlling with quasi-optimal convergence rate and another AMFEM for controlling without convergence rate. However, we are not aware of any existing AMFEM for the Hodge Laplace equation for controlling the error in the natural mixed variational norm.

On the other hand, the authors in [12, 13] developed the pseudostress-velocity formulation (5.4) for the Stokes equation, which can be numerically solved by the classical RT and BDM element mixed methods. Let denote the pseudostress, the velocity, and the finite element solutions produced by some AMFEM. Following the analysis of AMFEMs for Poisson’s equation, [15, 28] recently developed quasi-optimal AMFEMs for the pseudostress-velocity formulation that control the error , where is a positive semi-definite operator given in (5.3). Since is only a semi-norm, incorporation of is necessary for achieving norm convergence. Unlike Poisson’s equation, the velocity field in fluids is clearly an important physical quantity. From this perspective, an AMFEM for controlling is favorable.

Motivated by the Hodge Laplace and Stokes equations, this paper is devoted to the quasi-optimal adaptive mixed method for controlling the natural norm error. To this end, we consider the generalized Hodge Laplace equation (2.5), which covers the mixed formulation of Poisson’s eqaution and the pseudostress-velocity formulation of the Stokes equation. (2.5) also covers the mixed formulation of the Hodge Laplace equation with index provided the -th cohomology group vanishes, e.g., is simply connected when . In Section 5, we will restate our results in the classical context. The contribution of this paper is as follows.

1. Using the Demlow–Hirani regular decomposition and Falk–Winther cochain projection in FEEC, we prove the quasi-optimality of the adptive algorithm AMFEM for reducing the error in the -norm for the generalized Hodge Laplace equation. In particular, we obtain quasi-optimal AMFEMs for the Hodge Laplace equation. In the special case , i.e., Poisson’s equation, we obtain an AMFEM that reduces the error in the -norm.

2. By posing the Stokes equation on the de Rham complex of vector-valued differential forms, we modify the aforementioned tools in FEEC to derive a reliable and efficient a posteriori error estimator, and the first quasi-optimal AMFEM for the Stokes equation that reduces the error . The authors in [16] used to compute a more accurate postprocessed approximation and derived an error estimator for However, the reliability of such estimator depends on the -regularity of , e.g., is convex.

3. Our results for the Poisson and Stokes equations hold on general Lipschitz polyhedral domain . In contrast, existing analysis of AMFEMs in e.g., [18, 8, 15, 28] assumes that the Helmholtz decomposition contains no harmonic vector fields, which hinges on the topology of the domain , e.g., the -th Betti number of is .

An important ingredient of our convergence analysis is the quasi-orthogonality in Theorem 4.5. We observe that the -regular decomposition in [22] yields compact operators in Corollary 3.3 and develop a weak convergence result in Theorem 4.2 for the Petrov-Galerkin method. Note that map weakly convergent sequences to strongly convergent ones. Using this fact and the -bounded smoothed projection in [20], we obtain the quasi-orthogonality between and . Combining it with the quasi-orthogonality between and obtained in [30], the quasi-optimal convergence rate follows with a somehow standard procedure using the idea of estimator reduction, see [25]. Feischl et al. first used the weak convergence technique to prove quasi-optimal convergence rate of AFEMs in [25], where they observed that the lower order terms in order elliptic equation are compact perturbations. As far as we know, there is no convergence analysis of adaptive mixed methods in literature based on the weak convergence technique.

The rest of this paper is organized as follows. In Section 2, we introduce the closed Hilbert complex, de Rham complex, and the generalized Hodge Laplace equation. In Section 3, we derive reliable and efficient a posteriori error estimator for the generalized Hodge Laplace equation on the de Rham complex. Section 4 is devoted to the convergence and optimality analysis of the algorithm AMFEM. In Section 5, we use previous results and correspondence between functions and differential forms to obtain results on scalar Poisson, vector Poisson, and Stokes equations.

## 2. Hilbert complex and de Rham complex

Following the convention of [3, 4], we introduce FEEC in this section.

### 2.1. Hilbert complex and approximation

Given Hilbert spaces , we say is a closed, densely-defined operator if the domain is a dense subspace of , is linear, and the graph is a closed subset of . Let denote the inner product on , . The adjoint operator is defined to be the operator whose domain is

 D(T∗)={v∈X2:∃w∈X1, such that ⟨Tu,v⟩X2=⟨u,w⟩X1 for all u∈D(T)},

in which case is also a densely-defined, closed operator. Let denote the range of , the kernel of , the closed range theorem holds:

 (2.1) R(T)⊥=N(T∗),

where denotes the orthogonal complement operation.

Consider the closed Hilbert complex

 ⋯→Wk−1dk−1−−−→Wkdk−→Wk+1dk+1−−−→⋯,

i.e., for each index , is a Hilbert space equipped with the inner product and norm , is a densely-defined, closed operator, is closed in , and . Let , , and denote the space of abstract harmonic forms. is also called the -th cohomology group since . The cochain complex has the domain complex as a subcomplex:

 ⋯→Vk−1dk−1−−−→Vkdk−→Vk+1dk+1−−−→⋯.

Here is the domain of , equipped with the -inner product

 ⟨u,v⟩V:=⟨u,v⟩+⟨dku,dkv⟩,

and corresponding -norm Let There exists a constant such that

 (2.2) ∥v∥V≤cP∥dkv∥ for all v∈Zk⊥V.

In FEEC literature, (2.2) is called the Poincaré inequality.

For each index , choose a finite-dimensional subspace of . We assume that so that is a subcomplex of . Let be the same space but equipped with the -inner product . Similarly to the continuous case, let and . Note that in general and . In order to derive a posteriori error estimate on , we assume the existence of a bounded cochain projection from to . To be precise, for each index , maps onto , id, , and is uniformly bounded with respect to the discretization parameter . It has been shown in [4] that the discrete Poincaré inequality holds:

 ∥v∥V≤cP∥πkℓ∥V∥dkv∥ for all v∈Zk⊥ℓ.

### 2.2. Generalized Hodge Laplacian and approximation

For each index , let denote the adjoint operator of and . Throughout the rest of this paper, we may drop the superscript or subscript provided no confusion arises. On the closed Hilbert complex , Arnold, Falk and Winther [4] considered the abstract Hodge Laplace equation

 (dd∗+d∗d)u=f,

where and satisfy the compatibility condition Note that and it is difficult to construct finite element subspaces of . Therefore, the authors in [4] considered the mixed formulation: Find , such that

 (2.3) ⟨σ,τ⟩−⟨dτ,u⟩ =0, τ∈Vk−1, ⟨dσ,v⟩+⟨du,dv⟩+⟨v,p⟩ =⟨f,v⟩, v∈Vk, ⟨u,q⟩ =0, q∈Hk.

Using the discrete complex , the mixed method for (2.3) seeks such that

 (2.4) ⟨σℓ,τ⟩−⟨dτ,uℓ⟩ =0, τ∈Vk−1ℓ, ⟨dσℓ,v⟩+⟨duℓ,dv⟩+⟨v,pℓ⟩ =⟨f,v⟩, v∈Vkℓ, ⟨uℓ,q⟩ =0, q∈Hkℓ.

In general, and (2.4) is a nonconforming method . For the sake of simplicity, we assume the -th cohomology group and consider the generalized Hodge Laplacian problem: Find , such that

 (2.5) ⟨Cσ,τ⟩−⟨dτ,u⟩ =0, τ∈Vk−1, ⟨dσ,v⟩+⟨du,dv⟩ =⟨f,v⟩, v∈Vk,

where is a self-adjoint, positive semi-definite, continuous linear operator. is not necessarily positive definite with respect to and may not be define a norm on . We assume that there exists a constant with

 (2.6) C−1C∥τ∥2V≤⟨Cτ,τ⟩+∥dτ∥2≤CC∥τ∥2V

for all . For , let

 ⟨τ1,τ2⟩VC:=⟨Cτ1,τ2⟩+⟨dτ1,dτ2⟩.

(2.6) shows that is an inner product on and the -norm is equivalent to The mixed method for solving (2.5) is to find satisfying

 (2.7a) ⟨Cσℓ,τ⟩−⟨dτ,uℓ⟩ =0,τ∈Vk−1ℓ, (2.7b) ⟨dσ,v⟩+⟨duℓ,dv⟩ =⟨f,v⟩,v∈Vkℓ.

Thanks to the cochain projection we obtain and the well-posedness of (2.5) and (2.7), see Theorem 2.1. Assuming and using (2.7), we obtain the Galerkin orthogonality

 (2.8a) ⟨C(σℓ+1−σℓ),τ⟩−⟨dτ,uℓ+1−uℓ⟩ =0,τ∈Vk−1ℓ, (2.8b) ⟨d(σℓ+1−σℓ),v⟩+⟨d(uℓ+1−uℓ),dv⟩ =0,v∈Vkℓ.

Let . The next theorem shows that satisfies the continuous and discrete inf-sup condition, which implies the well-posedness of (2.5) and (2.7). The proof is the same as Theorem 3.2 in [4].

###### Theorem 2.1.

Assume . There exists a constant depending only on , such that

 γ(∥ξ∥VC+∥w∥V)≤supτ∈Vk−1,v∈VkB(ξ,w;τ,v)∥τ∥VC+∥v∥V,

for all . In addition, there exists a constant depending only on , such that

 γℓ(∥ξℓ∥VC+∥wℓ∥V)≤supτ∈Vk−1ℓ,v∈VkℓB(ξℓ,wℓ;τ,v)∥τ∥VC+∥v∥V,

for all .

Demlow and Hirani [22] used the continuous inf-sup condition to derive their error estimator for the method (2.4). Using the discrete inf-sup condition, we obtain the discrete upper bound of the abstract natural norm error.

###### Lemma 2.2.

For it holds that

 γℓ+1(∥σℓ+1−σℓ∥VC+∥uℓ+1−uℓ∥V) ≤supτ∈Vk−1ℓ+1,∥τ∥VC=1{⟨Cσℓ,τ−πℓτ⟩−⟨d(τ−πℓτ),uℓ⟩} +supv∈Vkℓ+1,∥v∥V=1{⟨f−dσℓ,v−πℓv⟩−⟨duℓ,d(v−πℓv)⟩}.
###### Proof.

Let and . It follows from (2.8) and (2.7) that

 B(σℓ+1−σℓ,uℓ+1−uℓ;τ,v) =B(σℓ+1−σℓ,uℓ+1−uℓ;τ−πℓτ,v−πℓv) =⟨f,v−πℓv⟩−B(σℓ,uℓ;τ−πℓτ,v−πℓv).

Combining it with the discrete inf-sup condition in Theorem 2.1 completes the proof. ∎

### 2.3. De Rham complex and approximation

The de Rham complex is a canonical example of the closed Hilbert complex. Let be a bounded Lipschitz domain. For index , let denote the space of all smooth -forms which can be uniquely written as

 ω=∑1≤α1<⋯<αk≤nωαdxα1∧⋯∧dxαk,

where each is a multi-index, each coefficient and is the wedge product. For with the inner product of and is

 ⟨ω,η⟩:=∑1≤α1<⋯<αk≤n∫Ωωαηαdx.

The exterior derivative is given by

 (2.9) dω=∑1≤α1<⋯<αk≤nn∑j=1∂ωα∂xjdxj∧dxα1∧⋯∧dxαk.

Let denote the -norm given by and the space of -forms with -coefficients. Then can be understood in the distributional sense. Let . The following cochain complex

 L2Λ0(Ω)d→L2Λ1(Ω)d→⋯d→L2Λn−1(Ω)d→L2Λn(Ω)

is an example of the closed Hilbert complex . The -de Rham complex [corresponds to ()] is

 (2.10) HΛ0(Ω)d→HΛ1(Ω)d→⋯d→HΛn−1(Ω)d→HΛn(Ω).

In order to characterize the adjoint of , we need the Hodge star operator determined by for all The coderivative is then determined by . and are related by the integrating by parts formula

 (2.11) ⟨dω,μ⟩=⟨ω,δμ⟩+∫∂Ωtrω∧tr⋆μ,ω∈Λk(Ω),μ∈Λk+1(Ω),

where the trace operator on is the pullback for differential forms induced by the inclusion . If is replaced by in (2.11), denotes the trace on by abuse of notation. We make use of the spaces and . The next lemma characterizes the adjoint operator of .

###### Theorem 2.3 (Theorem 4.1 in [4]).

Let be the exterior derivative viewed as a densely-defined, closed operator with domain . Then the adjoint as a densely-defined, closed operator , has domain , and coincides with

The generalized Hodge Laplaican problem (2.5) on the de Rham complex uses . A more compact form of (2.5) is

 (2.12) Cσ=δu,dσ+δdu=f.

Since and , the boundary conditions , on are implicitly imposed in (2.12). When , (2.12) reduces to the standard Hodge Laplacian problem

 (2.13) (dδ+δd)u =f in Ω, tr⋆u=0,tr⋆du =0 on ∂Ω.

Let be a sequence of nested conforming simplicial triangulations of , where means is a refinement of . For , let denote the volume of and . We assume that is shape regular, namely,

 supℓ≥0maxT∈TℓrTρT≤CT0<∞,

where and are radii of circumscribed and inscribed spheres of the simplex , respectively. Let denote the space of -forms on with polynomial coefficients of degree . Let

 PrΛk(Tℓ)={v∈HΛk(Ω):v|T∈PrΛk(T) for all T∈Tℓ},

where is the interior product given by

 (2.14) (κω)x(v1,…,vk−1)=ωx(X(x),v1,…,vk−1),v1,…,vk−1∈Rn,

with being the position vector field. For let

 (2.15) Vk−1ℓ =Pr+1Λk−1(Tℓ) or P−r+1Λk−1(Tℓ), Vkℓ =P−r+1Λk(Tℓ) or PrΛk(Tℓ),

Other spaces with are chosen in the same way. In , there are different discrete subcomplexes on a simplicial triangulation .

## 3. A posteriori error estimate

In the rest of this paper, provided and is a generic constant depending only on and . Let be the space of -forms whose coefficients are in . Formula (2.11) still holds for and Let denote the norm with some depending on the context. Let denote the -inner product restricted to , and the -norms restricted to and , respectively.

To derive discrete upper bounds on the de Rham complex, we need the local -bounded cochain projection developed by Falk and Winther [24]. The existence of implies the discrete inf-sup condition by Theorem 2.1.

###### Theorem 3.1 (local V-bounded cochain projection).

For each index there exists a projection commuting with the exterior derivative, i.e.  and . For , let and . Then

 (3.1a) v−πkℓv =0 on T for v∈Vkℓ|DT, (3.1b) ∥πkℓv∥HΛk(T) ≲∥v∥HΛk(DT) for v∈HΛk(DT).

 (3.2) |πkℓv|H1(T)+h−1T∥v−πkℓv∥T+h−12T∥tr(v−πkℓv)∥∂T≲|v|H1(DT).
###### Proof.

Properties (3.1a) and (3.1b) are given by Falk and Winther in [24]. Let be the interpolation given by applying the Scott–Zhang interpolation (cf. [41]) on to each coefficient of It follows from the property (3.1a) and the same property of that

 πkℓv−Iℓv=0 on T for v∈Vkℓ|DT,

and thus

 (3.3) ∥πkℓv−Iℓv∥T≲hT|v|H1(DT)

by the Bramble–Hilbert lemma. In addition, it is well-known that

 (3.4) |Iℓv|H1(T) ≲|v|H1(DT),∥v−Iℓv∥T≲hT|v|H1(DT).

Then using the triangle inequality, (3.3) and (3.4), we obtain

 |πkℓv|H1(T)≤|πkℓv−Iℓv|H1(T)+|Iℓv|H1(T) ≲|v|H1(DT), ∥v−πkℓv∥T≤∥v−Iℓv∥T+∥Iℓv−πkℓv∥T ≲hT|v|H1(DT).

Combining it with the trace inequality verifies

Using (3.2) and the bounded overlapping property of , we obtain

 (3.5) |πℓv|2H1(Ω)+∑T∈Tℓh−2T∥v−πℓv∥2T+h−1T∥tr(v−πℓv)∥2∂T≲|v|2H1(Ω).

In addition to , we need an -regular decomposition result, see Lemma 5 in [22]. The proof therein hinges on the technical -solution regularity of the equation under the Dirichlet boundary condition, see e.g., [31, 40]. In our convergence analysis, the linearity of such regular decomposition is also required. Since only the natural boundary condition is considered, we give a simple proof of the regular decomposition below. For convenience, let

###### Theorem 3.2 (regular decomposition).

Let be a bounded Lipschitz domain in For , there exist bounded linear operators and , such that for ,

 v=dKk1v+Kk2v.
###### Proof.

When and When , is identified with the divergence operator , see Section 5. In this case, let and be the -regular right inverse of i.e.,