On a mixed FEM and a FOSLS with H-1 loads

2025-05-02 0 0 662.79KB 20 页 10玖币

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ON A MIXED FEM AND A FOSLS WITH H−1LOADS

THOMAS FÜHRER

Abstract. We study variants of the mixed ﬁnite element method (mixed FEM) and the

ﬁrst-order system least-squares ﬁnite element (FOSLS) for the Poisson problem where we

replace the load by a suitable regularization which permits to use H−1loads. We prove that

any bounded H−1projector onto piecewise constants can be used to deﬁne the regularization

and yields quasi-optimality of the lowest-order mixed FEM resp. FOSLS in weaker norms.

Examples for the construction of such projectors are given. One is based on the adjoint of a

weighted Clément quasi-interpolator. We prove that this Clément operator has second-order

approximation properties. For the modiﬁed mixed method we show optimal convergence

rates of a postprocessed solution under minimal regularity assumptions — a result not valid

for the lowest-order mixed FEM without regularization. Numerical examples conclude this

work.

1. Introduction

In this work we study a mixed ﬁnite element method (FEM) and a ﬁrst-order least-squares

FEM (FOSLS) for the Poisson problem with H−1(Ω) loads where H−1(Ω) denotes the topo-

logical dual of the Sobolev space H1

0(Ω) and Ω⊂Rn(n= 2,3) denotes a bounded Lipschitz

domain with polytopal boundary. Both numerical methods are based on the following ﬁrst-

order reformulation of the Poisson problem with homogeneous Dirichlet boundary conditions,

div σ=−fin Ω,(1a)

σ− ∇u= 0 in Ω,(1b)

u= 0 on Γ := ∂Ω.(1c)

Given f∈L2(Ω) the mixed FEM seeks (uT,σT)∈WT×ΣTsuch that

(σT,τ)+(uT,div τ) = 0,(2a)

(div σT, v) = (−f , v)(2b)

for all (v, τ)∈WT×ΣT:= P0(T)× RT 0(T), where Tdenotes a shape-regular mesh

of simplices of Ω,Pp(T)denotes the space of T-piecewise polynomials of degree less or

equal to p∈N0and RT 0(T)is the lowest-order Raviart–Thomas space. Given f∈L2(Ω)

the FOSLS seeks the minimizer of the L2(Ω) residuals of (1a)–(1b) over the discrete space

UT×ΣT=P1(T)∩H1

0(Ω) ×ΣT, i.e.,

(uT,σT) = arg min

(v,τ)∈UT×ΣT

kdiv τ+fk2+k∇v−τk2.(3)

Date: February 3, 2023.

2010 Mathematics Subject Classiﬁcation. 65N30, 65N12 .

Key words and phrases. least-squares method, mixed FEM, singular data.

Acknowledgment. This work was supported by ANID through FONDECYT project 1210391.

arXiv:2210.14063v2 [math.NA] 1 Feb 2023

Both methods, (2) and (3), are not well deﬁned for f∈H−1(Ω). In the recent article [15]

we proposed to replace the load in (3) by a suitable polynomial approximation. The very

same ideas in the analysis can be applied for the mixed FEM (2). To describe the new

variants, let Q?

T:H−1(Ω) → P0(T)⊆H−1(Ω) denote a bounded projection operator, i.e.,

Tφ=φfor all φ∈ P0(T)and kQ?

Tφk−1.kφk−1for all φ∈H−1(Ω).

The modiﬁed methods are deﬁned as follows:

Modiﬁed mixed FEM: Given f∈H−1(Ω) seek (uT,σT)∈WT×ΣTsuch that

(σT,τ)+(uT,div τ) = 0,(4a)

(div σT, v) = (−Q?

Tf , v)(4b)

for all (v, τ)∈WT×ΣT.

Modiﬁed FOSLS: Given f∈H−1(Ω) solve

(uT,σT) = arg min

(v,τ)∈UT×ΣT

kdiv τ+Q?

Tfk2+k∇v−τk2.(5)

In our recent work [15] we proved that the solution (uT,σT)of (5) satisﬁes the error

estimate

ku−uTk1+kσ−σTk.hskfk−1+s

where s∈[0,1] depends on Ωand the regularity of f, and his the maximum element

diameter. A similar estimate may be derived for the solution of (4) following the techniques

from [15], or the ones presented here, see Corollary 5 below.

In the article at hand we complement on our results from [15] in that we show quasi-

optimality of both the modiﬁed methods (4) and (5): Let (u, σ)∈H1

0(Ω)×L2(Ω; Rd)denote

the solution of (1). If (uT,σT)∈WT×ΣTdenotes the solution of (4), then (see Theorem 3)

ku−uTk+kσ−σTk.min

(v,τ)∈WT×ΣT

ku−vk+kσ−τk,

or if (uT,σT)∈UT×ΣTdenotes the solution of (5), then (see Theorem 6)

ku−uTk1+kσ−σTk.min

(v,τ)∈UT×ΣT

ku−vk1+kσ−τk.

The mixed FEM (2) and FOSLS (3) have been studied thoroughly and we refer the inter-

ested reader to [3, 16, 2] for an introduction, overview and further literature on these meth-

ods. A variant of the hybrid higher-order method (known as HHO) with H−1(Ω) loads is

introduced and analyzed in [12]. For further details on and constructions of diﬀerent H−1(Ω)

projection operators onto piecewise polynomial spaces we refer to the recent work [10] where

also various applications are discussed. In [19] a general theory for the approximation of

rough linear functionals is developed.

Postprocessing schemes for the mixed method (2) are well known [22], and optimal con-

vergence rates for higher-order elements can be shown, whereas the lowest-order case, as

considered here, requires suﬃciently regular solutions, see, e.g., [22, Theorem 2.1 and Re-

mark 2.1]. In the work at hand we prove that the postprocessing scheme from [22] applied to

solutions of the modiﬁed mixed FEM (4) yields optimal rates with only minimal regularity

assumptions.

The analysis of the latter is based on the dual of a weighted Clément quasi-interpolator.

The advantage of our proposed construction is that the Clément operator reconstructs an

approximation with second-order approximation properties from an elementwise projection

on constants. For an overview on Clément quasi-interpolators we refer to the works [8, 6] and

for additional information and applications to [7]. As a side product of our analysis we obtain

a result on the approximation by piecewise constants in the dual space of H2(Ω) ∩H1

0(Ω)

(Corollary 13).

Our results on quasi-optimality in weaker norms might also be of interest for the analysis

of FOSLS for eigenvalue problems [1]. The authors of [21] deﬁne a superconvergent FEM

based on the postprocessing technique from [22]. Our new ﬁndings for the postprocessing

scheme (Section 4) could also improve the results from [21] for the lowest-order case.

In this article we only consider lowest-order discretizations, though, many results can be

extended to the higher-order case. E.g., for the FOSLS we refer the reader to the very recent

work [20, Remark 4.7]. We restrict the presentation to n= 2,3but note that our results

are valid for n= 1. The remainder of this work is organized as follows: Section 2 introduces

some notation and contains the statement and proofs of the quasi-optimality results stated

above. In Section 3 we study a weighted Clément quasi-interpolator and discuss some of its

main properties. Optimal error estimates for the postprocessed solution of (4) and optimal

L2(Ω) error estimates for the scalar solution of (5) are given in Section 4. This article closes

with various numerical experiments (Section 5).

2. Quasi-optimality

This section is devoted to the proof of quasi-optimality results of the modiﬁed variants (4)

and (5) of the mixed FEM and FOSLS claimed in the introduction. Before we give details in

Section 2.3 we recall some known properties of projection operators needed for the analysis

in Section 2.2. The proof of quasi-optimality requires a H−1(Ω) bounded projection operator

and we also give an example of such an operator that is easy to implement.

2.1. Sobolev spaces. For a Lipschitz domain K⊆Ωwe denote by Hk(K),Hk

0(K),k∈N

the usual Sobolev spaces with norms k · kK,k. If K= Ω we simply write k · kk. The

space H1

0(Ω) is equipped with the norm k·k1:= k∇(·)kwhere k·k is the L2(Ω) norm

with inner product (·,·). Similarly, k·kKis the L2(K)norm with inner product (·,·)K.

Intermediate Sobolev spaces with index sare deﬁned by (real) interpolation, e.g., e

Hs(Ω) =

[L2(Ω), H1

0(Ω)]s,Hs(Ω) = [L2(Ω), H1(Ω)]swith norm denoted by k · ks. Dual spaces of

Sobolev spaces are understood with respect to the extended L2inner product, e.g, the dual

of H1

0(Ω) is denoted by H−1(Ω) and equipped with the dual norm

kφk−1= sup

06=v∈H1

0(Ω)

(φ, v)

kvk1

Note that H−s(Ω) = ( e

Hs(Ω))0with norms k·k−s,s∈(0,1).

2.2. Projection and interpolation operators. Let Tdenote a regular mesh of simplices

of Ωwith hT∈L∞(Ω) denoting the elementwise mesh-size function, hT|T= diam(T)for

T∈ T . With Vwe denote all vertices of Tand V0=V \ Γare the interior vertices. The set

of n+ 1 vertices of an element T∈ T is VT. The patch of all elements of Tsharing a node

z∈ V is denoted by ωzand Ωzis used for the domain associated to ωz. The element patch

ωTis the union of all vertex patches ωzwith z∈ VTand ΩTis the corresponding domain.

Let Π0

T:L2(Ω) →WT=P0(T)denote the L2(Ω)-orthogonal projection which has the

ﬁrst-order approximation property

k(1 −Π0

T)φkT.hTk∇φkTfor all φ∈H1(T), T ∈ T .

Here, and in the remainder, the notation A.Bmeans that there exists a generic constant,

possibly depending on the shape-regularity constant κTand Ω, such that A≤C·B. The

notation AhBmeans A.Band B.A. The shape-regularity constant of a mesh Tis

given by

κT= max

T∈T

|T|,

where |·|denotes the volume measure.

Recall that ΣT=RT 0(T)is the lowest-order Raviart–Thomas space. We denote by

ΠRT

T:H(div; Ω) →ΣTthe projector constructed in [11]. It has the following properties,

see [11, Theorem 3.2], where RT 0(T)denotes the lowest-order Raviart–Thomas space on

the element T.

div ΠRT

Tσ= Π0

Tdiv σand(6a)

kσ−ΠRT

Tσk2.X

T∈T

min

τ∈RT 0(T)kσ−τk2

T+khT(1 −Π0

T) div σk2

(6b)

and, in particular,

kΠRT

Tσk2

H(div;Ω) := kΠRT

Tσk2+kdiv ΠRT

Tσk2.kσk2

H(div;Ω)

(6c)

for all σ∈H(div; Ω).

There are several possibilities to construct a bounded projection Q?

T:H−1(Ω) → P0(T).

We refer the interested reader to [10] for an overview on existing operators and the construc-

tion of a family of H−1(Ω) projectors into polynomial spaces. Here, we follow the construc-

tion presented in [15] resp. [14, Section 2.4]. First, deﬁne the averaged Scott–Zhang-type

quasi-interpolator JSZ

T:H1

0(Ω) → P1(T)∩H1

0(T)by

JSZ

Tv=X

z∈V0

(v , ψz)ηz,

where {ηz}z∈V0is the nodal basis of P1(T)∩H1

0(Ω), i.e., ηz(z0) = δz,z0for all z, z0∈ V0, and

δz,z0denotes the Kronecker-δsymbol. Furthermore, ψz∈ P1(T)with supp ψz⊆Ωzdenotes

the bi-orthogonal dual basis function satisfying

(ψz, ηz0) = δz,z0∀z, z0∈ V0.

An explicit representation is given by

ψz|Ωz=1

|Ωz|(n+ 1)(n+ 2)ηz−(n+ 1).

For T∈ T deﬁne the bubble function ηb,T := cTQz∈VTηzwith cTchosen so that (ηb,T ,1) = 1

and

BTv=X

T∈T

(v , χT)ηb,T ,

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摘要：

ONAMIXEDFEMANDAFOSLSWITHH1LOADSTHOMASFÜHRERAbstract.Westudyvariantsofthemixedniteelementmethod(mixedFEM)andtherst-ordersystemleast-squaresniteelement(FOSLS)forthePoissonproblemwherewereplacetheloadbyasuitableregularizationwhichpermitstouseH1loads.WeprovethatanyboundedH1projectorontopiecewiseconst...

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On a mixed FEM and a FOSLS with H-1 loads

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