We present a numerical approximation method for linear diffusion-reaction
problems with possibly discontinuous Dirichlet boundary conditions. The
solution of such problems can be represented as a linear combination of
explicitly known singular functions as well as of an $H^2$-regular part. The
latter part is expressed in terms of an elliptic problem with regularized
Dirichlet boundary conditions, and can be approximated by means of a Nitsche
finite element approach. The discrete solution of the original problem is then
defined by adding the singular part of the exact solution to the Nitsche
approximation. In this way, the discrete solution can be shown to converge of
second order with respect to the mesh size

Baumann, Ramona

Wihler, Thomas P.

Computational Methods in Applied Mathematics

English

arXiv

Abstract
               We present a numerical approximation method for linear elliptic diffusion-reaction problems with possibly discontinuous Dirichlet boundary conditions. The solution of such problems can be represented as a linear combination of explicitly known singular functions as well as of an 
                     
                        
                           
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                  -regular part. The latter part is expressed in terms of an elliptic problem with regularized Dirichlet boundary conditions, and can be approximated by means of a Nitsche finite element approach. The discrete solution of the original problem is then defined by adding back the singular part of the exact solution to the Nitsche approximation. In this way, the discrete solution can be shown to converge of second order in the 
                     
                        
                           
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                  -norm with respect to the mesh size.</jats:p

Ramona Baumann

Thomas P. Wihler

Crossref

A Nitsche Finite Element Approach for Elliptic Problems with Discontinuous Dirichlet Boundary Conditions

We present a numerical approximation method for linear elliptic diffusion-reaction problems with possibly discontinuous Dirichlet boundary conditions. The solution of such problems can be represented as a linear combination of explicitly known singular functions as well as of an H²-regular part. The latter part is expressed in terms of an elliptic problem with regularized Dirichlet boundary conditions, and can be approximated by means of a Nitsche finite element approach. The discrete solution of the original problem is then defined by adding back the singular part of the exact solution to the Nitsche approximation. In this way, the discrete solution can be shown to converge of second order in the L²-norm with respect to the mesh size

Wihler, Thomas

Bern Open Repository and Information System (BORIS)

source: https://doi.org/10.7892/boris.125542 | downloaded: 29.9.2023A NITSCHE FINITE ELEMENT APPROACH FOR ELLIPTICPROBLEMS WITH DISCONTINUOUS DIRICHLET BOUNDARYCONDITIONSRAMONA BAUMANN AND THOMAS P. WIHLERAbstract. We present a numerical approximation method for linear diffusion-reaction problems with possibly discontinuous Dirichlet boundary conditions.The solution of such problems can be represented as a linear combination ofexplicitly known singular functions as well as of an H2-regular part. The latterpart is expressed in terms of an elliptic problem with regularized Dirichletboundary conditions, and can be approximated by means of a Nitsche finiteelement approach. The discrete solution of the original problem is then definedby adding the singular part of the exact solution to the Nitsche approximation.In this way, the discrete solution can be shown to converge of second orderwith respect to the mesh size.1. IntroductionGiven a bounded, open and convex polygonal domain Ω ⊂ R2 with straightedges, we consider the linear diffusion-reaction problem−∆u+ µu = f in Ω,(1)u = g on Γ,(2)where Γ = ∂Ω denotes the boundary of Ω, µ ∈ L∞(Ω) is a nonnegative function,f ∈ L2(Ω) is a source term, and g ∈ L2(∂Ω) is a possibly discontinuous functionon Γ whose precise regularity will be specified later on.Various formulations for (1)–(2), where the Dirichlet boundary data does notnecessarily belong toH1/2(Γ), exist in the literature. For instance, the very weak for-mulation is based on twofold integration by parts of (1) and, thereby, incorporatesthe Dirichlet boundary conditions in a natural way. It seeks a solution u ∈ L2(Ω)such that−∫Ωu∆v dx +∫Ωµuv dx =∫Ωfv dx−∫Γg∇v · n dsfor any v ∈ H2(Ω) ∩H10 (Ω), where we write n for the unit outward normal vectorto the boundary Γ. Alternatively, the following saddle point formulation, whichtraces back to the work [9], may be applied: provided that g ∈ H1/2−ε(∂Ω), forsome ε ∈ [0, 1/2), find u ∈ H1−ε(Ω) with u|Γ = g such that(3)∫Ω∇u · ∇v dx +∫Ωµuv dx =∫Ωfv dxMathematics Institute, University of Bern, SwitzerlandE-mail address: wihler@math.unibe.ch.2010 Mathematics Subject Classification. 65N30.Key words and phrases. Second-order elliptic PDE, discontinuous Dirichlet boundary condi-tions, Nitsche FEM.1arXiv:1707.00764v1  [math.NA]  3 Jul 20172 R. BAUMANN AND T. P. WIHLERfor all v ∈ H1+ε(Ω)∩H10 (Ω); for results dealing with finite element approximationsof (3), we refer to [4]. Another related approach is based on weighted Sobolev spaces(accounting for the local singularities of solutions with discontinuous boundarydata), and has been analyzed in the context of hp-type discontinuous Galerkinmethods in [7].The main idea of this paper is to represent the (weak) solution of (1)–(2) in termsof a regular H2 part as well as an explicitly known singular part (Section 2.4). Thelatter is expressed by means of suitable singular functions which account for thelocal discontinuities in the Dirichlet boundary data (Section 2.2). Here, it is crucialto ensure that the boundary data of the regular problem is sufficiently smoothas to provide an H2 trace lifting (see Section 2.3). We shall employ a classicalNitsche technique in order to discretize the regular part of the solution, and definethe numerical approximation of (1)–(2) by adding back the (exact) singular part(Section 3.2). A numerical experiment (Section 3.3) underlines that our approachprovides optimally converging results.Throughout the paper we shall use the following notation: For an open do-main D ⊂ Rn, n ∈ {1, 2}, and p ∈ [1,∞], we denote by Lp(D) the class of Lebesguespaces on D. For p = 2, we write ‖ · ‖0,D to signify the L2-norm on D. Further-more, for an integer k ∈ N0, we let Hk(D) be the usual Sobolev space of order kon D, with norm ‖ · ‖k,D and semi-norm | · |k,D. The set H10 (D) represents thesubspace of H1(D) of all functions with zero trace along ∂D. If D is representedas a (disjoint) finite union of open sets, that is, D =⋃iDi, and X is any class offunction spaces, then we write Xpw(D) = ΠiX(Di) to mean the set of all functionswhich belong piecewise (with respect to the partition {Di}i) to X.2. Problem formulationThe aim of this section is to establish a suitable framework for the weak solutionof (1)–(2).2.1. Notation. Let A = {Ai}Mi=1 ⊂ ∂Ω, with Ai 6= Aj , for 1 ≤ i 6= j ≤ M , be afinite set of points on the boundary of the polygonal domain Ω, which are numberedin counter-clockwise direction along ∂Ω; the points in A mark the locations wherethe Dirichlet boundary condition g from (2) exhibits discontinuities. Furthermore,we denote by Γi ⊂ Γ, i = 1, 2, . . . ,M , the open edge which connects the twopoints Ai and Ai+1; in the sequel, we shall identify indices 0 'M , 1 'M +1, etc.;for instance, we have AM+1 = A1 and A0 = AM , or ΓM+1 = Γ1 and Γ0 = ΓM ,etc. Moreover, let ωi ∈ (0, π] signify the interior angle of Ω at Ai (in counter-clockwise direction). Finally, for φ ∈ C0pw(Γ), i.e., φ|Γi ∈ C0(Γi), for 1 ≤ i ≤M , weset φi := φ|Γi , and define the one-sided limitsφ(A+i ) = limx→Aix∈Γiφi(x), φ(A−i ) = limx→Aix∈Γi−1φi−1(x),and the jumps [[φ]]i = φ(A+i )− φ(A−i ), for i = 1, . . . ,M .2.2. Singular functions. In the following, based on the partition Γ =⋃Mi=1 Γi,we assume that the boundary data g from (2) satisfies(4) g ∈ H2pw(Γ),A NITSCHE FEM FOR PROBLEMS WITH DISCONTINUOUS BOUNDARY CONDITIONS 3i.e., with the notation above, we have gi ∈ H2(Γi), for 1 ≤ i ≤ M . We note thecontinuous Sobolev embedding H1(Γi) ↪→ L∞(Γi), i.e.,(5) supx∈Γi|v(x)| ≤ C‖v‖1,Γi , ∀v ∈ H1(Γi), i = 1, . . . ,M,for a constant C = C(Γi) > 0. In particular, this implies that the values of g(A±i )and g′(A±i ), with g′ denoting the (edgewise) tangential derivative of g in counter-clockwise direction along Γ, are well-defined. Hence, for ri 6= 0, we may considerthe singular functions (cf. [8, Lemma 6.1.1]), for 1 ≤ i ≤M ,(6) Θi(ri, θi) =g(A+i )−θiωi[[g]]i if ωi ∈ (0, π),g(A+i )−1π(θi[[g]]i + σi(ri, θi)[[g′]]i) if ωi = π,withσi(ri, θi) = ri (ln(ri) sin(θi) + θi cos(θi)) .Here, (ri, θi) denote polar coordinates with respect to a local coordinate systemcentered at Ai such that θi = 0 on Γi, and θi = ωi on Γi−1. We note that Θi isharmonic away from Ai, i.e., ∆Θi = 0 in Ω. Since Θi is smooth away from Ai,there holds(7) [[Θi]]j = δij [[g]]i, 1 ≤ i, j ≤M,where δij is Kronecker’s delta. In addition, for ωj = π, we have(8) [[Θ′i]]j = δij [[g′]]j ,for i = 1, . . . ,M .2.3. Trace lifting. Defining the function(9) ĝ : Γ→ R, ĝ := g −M∑i=1Θi|Γ,with Θi from (6), and recalling (7), we note that(10) [[ĝ]]j = [[g]]j −M∑i=1[[Θi]]j = 0, 1 ≤ j ≤M,i.e., ĝ is continuous along the boundary Γ. Similarly, whenever ωj = π, using (8),we have(11) [[ĝ′]]j = [[g′]]j −M∑i=1[[Θ′i]]j = 0.Lemma 2.1. There holds the estimateM∑i=1‖ĝi‖2,Γi ≤ CM∑i=1‖gi‖2,Γi ,where C > 0 is a constant independent of g.4 R. BAUMANN AND T. P. WIHLERQixyΓiΓi−1AiAi−1Ai+1(x, y)Xi(x, y)Yi(x, y)Figure 1. Graphical illustration of (local) trace lifting construction.Proof. By definition of ĝ, see (9), for any 1 ≤ i ≤M , there holds‖ĝi‖2,Γi ≤ ‖gi‖2,Γi +M∑j=1‖Θj‖2,Γi .Since Θj is a linear function along both Γj−1 and Γj and smooth on⋃k 6=j−1,j Γk,we deduce the bound‖Θj‖2,Γi ≤ Cij(|g(A+j )|+ ω−1j |[[g]]j |+ |[[g′]]j |),where Cij > 0 is a constant depending on Aj and Γi. Hence,M∑i=1‖ĝi‖2,Γi ≤M∑i=1‖gi‖2,Γi +M∑i,j=1Cij(|g(A+j )|+ ω−1j |[[g]]j |+ |[[g′]]j |)≤M∑i=1‖gi‖2,Γi + CM∑i=1(‖gi‖∞,Γi + ‖g′‖∞,Γi) .Using (5), the proof is complete. The identities (10) and (11) together with the previous lemma imply the followingresult.Lemma 2.2. There exists a lifting Û ∈ H2(Ω) of the boundary data ĝ, i.e., Û |Γ = ĝin the sense of traces, with(12) ‖Û‖2,Ω ≤ CM∑i=1‖gi‖2,Γi ,where C > 0 is a constant independent of g.Proof. We use a partition of unity approach. Specifically, to each corner Ai of Ω,we associate a function φi ∈ C∞(Ω) such that∑Mi=1 φi(x) = 1 for any x ∈ Γ,and supp(φi) ∩ Γ ⊂ Γi−1 ∪ {Ai} ∪ Γi, for 1 ≤ i ≤M .Fix i ∈ {1, . . . ,M}. If 0 < ωi < π, we may assume, without loss of generality,that Ai coincides with the origin (0, 0), and the edge Γi can be placed on the firstA NITSCHE FEM FOR PROBLEMS WITH DISCONTINUOUS BOUNDARY CONDITIONS 5coordinate axis. Denoting the (Cartesian) coordinates in this system by (x, y), weletXi(x, y) =(x− ytan(ωi), 0), Yi(x, y) =(ytan(ωi), y);see Figure 1 for a graphical illustration. Observe thatXi|Γi−1 = 0, Xi|Γi = id,Yi|Γi−1 = id, Yi|Γi = 0,where id is the identity function. Then, for (x, y) ∈ Ω, we define the liftingÛi ={(ĝ|Γi ◦Xi + ĝ|Γi−1 ◦ Yi − ĝ(Ai))φi in Qi ∩ Ω,0 on Ω \ Qi,whereQi = {x = ω1(Ai+1 −Ai) + ω2(Ai−1 −Ai) : ω1, ω2 ∈ (0, 1)} ;cf. the gray area in Figure 1. The lifting Ûi satisfies the boundary condition(13) Ûi|Γ = ĝφi|Γ.Furthermore, we note that‖Ûi‖2,Ω ≤ C(‖ĝi−1‖2,Γi−1 + ‖ĝi‖2,Γi + |ĝ(Ai)|).Using (5), we obtain(14) ‖Ûi‖2,Ω ≤ C(‖ĝi−1‖2,Γi−1 + ‖ĝi‖2,Γi).If ωi = π, then the function ĝφi|Γ belongs to H3/2(Γ), and by the trace theorem,there exists Ûi ∈ H2(Ω) which again satisfies (13) as well as (14). Therefore, lettingÛ =M∑i=1Ûi,we see that Û |Γ = ĝ, and‖Û‖2,Ω ≤M∑i=1‖Ûi‖2,Ω ≤ CM∑i=1‖ĝi‖2,Γi .Employing Lemma 2.1 completes the argument. 2.4. Weak solution. Let(15) f̂ := f − µM∑i=1Θi ∈ L2(Ω).Then, proceeding analogously as in the proof of Lemma 2.1, we deduce that(16) ‖f̂‖0,Ω ≤ ‖f‖0,Ω + µM∑i=1‖Θi‖0,Ω ≤ ‖f‖0,Ω + CM∑i=1‖gi‖2,Γi ,with a constant independent of f and g. Consider the regularized problem−∆û+ µû = f̂ in Ω,(17)û = ĝ on Γ,(18)where ĝ is the boundary function from (9).6 R. BAUMANN AND T. P. WIHLERProposition 2.3. Let Ω be a convex and bounded polygonal domain. Then, thereexists a unique solution û ∈ H2(Ω) to (17)–(18) that satisfies the stability bound(19) ‖û‖2,Ω ≤ C(‖f‖0,Ω +M∑i=1‖gi‖2,Γi),with a constant C > 0 depending on Ω, and on µ.Proof. Proposition 2.2 provides the existence of a function Û ∈ H2(Ω) with Û∣∣Γ=ĝ. Since f̂+∆Û−µÛ belongs to L2(Ω), elliptic regularity theory in convex polygons(see, e.g., [2, 5, 6]) implies the existence of a unique remainder function ρ̂ ∈ H2(Ω)with−∆ρ̂+ µρ̂ = f̂ + ∆Û − µÛ in Ω,ρ̂ = 0 on Γ,and(20) ‖ρ̂‖2,Ω ≤ C‖f̂ + ∆Û − µÛ‖0,Ω ≤ C(‖Û‖2,Ω + ‖f̂‖0,Ω).Thus, the function û := Û + ρ̂ belongs to H2(Ω). Furthermore, it holds that−∆û+ µû = −∆Û + µÛ −∆ρ̂+ µρ̂ = f̂ in Ω,as well asû∣∣Γ= Û∣∣Γ+ ρ̂∣∣Γ= ĝ.In addition, combining (12) and (20) yields‖û‖2,Ω ≤ ‖Û‖2,Ω + ‖ρ̂‖2,Ω ≤ C(‖Û‖2,Ω + ‖f̂‖0,Ω)≤ C(M∑i=1‖gi‖2,Γi + ‖f̂‖0,Ω),which, by virtue of (16), results in the bound (19). Definition 2.4. We call the function u defined by(21) u := û+M∑i=1Θi,with û the unique H2-solution of (17)–(18), the weak solution of (1)–(2).Remark 2.5. It can be verified easily that the weak solution defined in (21) belongsto a class of weighted Sobolev spaces; cf., e.g., [2, 3]. The norms of these spacescontain local radial weights at the discontinuity points A of the Dirichlet boundarydata, and, thereby, account for possible singularities in the solution of (1)–(2).Based on an inf-sup theory, the work [7] shows that (1)–(2) exhibits a uniquesolution within this framework.3. Numerical approximationThe purpose of this section is to discretize (1)–(2) by a finite element approach.Specifically, we will employ a Nitsche method to obtain a numerical approximationof the elliptic problem (17), with the possibly non-homogeneous Dirichlet boundarycondition (18). The discrete solution will then be defined similarly as in (21).A NITSCHE FEM FOR PROBLEMS WITH DISCONTINUOUS BOUNDARY CONDITIONS 73.1. Meshes and spaces. We consider regular, quasi-uniform meshes Th of meshsize h > 0, which partition Ω ⊂ R2 into open disjoint triangles and/or parallelo-grams {K}K∈Th , i.e., Ω =⋃K∈Th K. Each element K ∈ Th is an affinely mappedimage of the reference triangle T̂ = {(x̂, ŷ) : −1 < x̂ < 1,−1 < ŷ < −x̂} or thereference square Ŝ = (−1, 1)2, respectively. Moreover, we define the conformingfinite element spaceV(Th) = {v ∈ H1(Ω) : v|K ∈ S(K),K ∈ Th},where, for K ∈ Th, we write S(K) to mean either the space P1(K) of all polynomialsof total degree at most 1 on K or the space Q1(K) of all polynomials of degree atmost 1 in each coordinate direction on K.3.2. Nitsche discretization. The classical Nitsche approach [10] for the numeri-cal approximation of (17)–(18) is given by finding ûh ∈ V(Th) such that(22) ah(ûh, v) = lh(v) for all v ∈ V(Th).Here, denoting by ∇h the elementwise gradient operator, we define the bilinearformah(w, v) =∫Ω{∇hw · ∇hv + µwv} dx−∫∂Ωv (∇hw · n) ds−∫∂Ωw (∇hv · n) ds+γh∫∂Ωwv ds,as well as the linear functionallh(v) =∫Ωf̂v dx−∫∂Ωĝ (∇hv · n) ds+γh∫∂Ωĝv ds,with ĝ and f̂ from (9) and (15), respectively. The penalty parameter γ > 0 appear-ing in both forms is chosen sufficiently large (but independent of the mesh size) asto guarantee the well-posedness of the weak formulation (22); this can be shown ina similar way as in the context of discontinuous Galerkin methods; see, e.g., [1]. Inaddition, referring to [10, Satz 2], cf. also [1, Section 5.1], there holds the a priorierror estimate(23) ‖û− ûh‖0,Ω ≤ Ch2|û|2,Ω,with a constant C = C(µ, f̂ , ĝ) > 0 independent of the mesh size h.Definition 3.1. Analogously to (21), we define the discrete solution of (1)–(2) by(24) uh := ûh +M∑i=1Θi,where ûh ∈ V(Th) is the Nitsche solution from (22), and {Θi}Mi=1 are the singularfunctions from (6).Theorem 3.2. Let u be the solution of (1)–(2) given by (21), and uh its discretecounterpart from (24). Then, there holds the a priori error estimate(25) ‖u− uh‖0,Ω ≤ Ch2,with a constant C = C(µ, f, g) > 0 independent of h.8 R. BAUMANN AND T. P. WIHLER10-3 10-2 10-1 10010-610-510-410-310-210-1100||u-uh||0,slope 2Figure 2. L2 error ‖u− uh‖0,Ω against mesh size h compared toa reference line with slope 2 (expected behaviour).Proof. We recall the definitions (21) and (24) in order to noticeu− uh = û+M∑i=1Θi −(ûh +M∑i=1Θi)= û− ûh.Therefore, applying (23) yields‖u− uh‖0,Ω ≤ Ch2|û|2,Ω.Finally, recalling (19) completes the proof. 3.3. Numerical example. On the rectangle Ω = (−1, 1)× (0, 1) we consider theelliptic boundary value problem−∆u+ u = e−r2(5− 4r2)θ in Ωu = g on Γ,with the Dirichlet boundary data g chosen such that the analytical solution is givenbyu(r, θ) = e−r2θ.Here, (r, θ) denote polar coordinates in R2. Note that the solution u is smoothalong Γ except at the origin, where it exhibits a discontinuity jump. In particular,it follows that u 6∈ H1(Ω).Starting from a regular coarse mesh, we investigate the practical performance ofthe a priori error estimate derived in Theorem 3.2 within a sequence of uniformlyrefined P1 elements. In Figure 2 we present a comparison of the L2 norm of theerror versus the mesh size h on a log-log scale for each of the meshes. Our results arein line with the a priori error estimate (25), and show that the discrete solution uhfrom (24) converges of second order with respect to the mesh size h. Moreover, inFigure 3 we show the Nitsche solution ûh ∈ V(Th) defined in (22), as well as thecomputed solution uh for a mesh consisting of 1024 elements.A NITSCHE FEM FOR PROBLEMS WITH DISCONTINUOUS BOUNDARY CONDITIONS 9-2.51-2-1.51-1-0.50.50.5000.5-0.50 -1011120.530.504-0.50 -1Figure 3. Nitsche solution (left) and discrete solution (right)based on a uniform mesh with 1024 elements.References1. D. N. Arnold, F. Brezzi, B. Cockburn, and L. D. Marini, Unified analysis of discontinuousGalerkin methods for elliptic problems, SIAM J. Numer. Anal. 39 (2001), 1749–1779.2. I. Babuška and B. Q. Guo, Regularity of the solution of elliptic problems with piecewiseanalytic data. I. Boundary value problems for linear elliptic equation of second order, SIAMJ. Math. Anal. 19 (1988), 172–203.3. I. Babuška and B. Q. Guo, Regularity of the solution of elliptic problems with piecewiseanalytic data. II. The trace spaces and application to the boundary value problems with non-homogeneous boundary conditions, SIAM J. Math. Anal. 20 (1989), no. 4, 763–781.4. I. Babuška, Error bounds for finite element method, Numer. Math. 16 (1971), no. 4, 322–333.5. M. Dauge, Elliptic boundary value problems on corner domains, Lecture Notes in Mathemat-ics, no. 1341, Springer-Verlag, 1988.6. P. Grisvard, Elliptic problems in nonsmooth domains, Monographs and Studies in Mathemat-ics, vol. 24, Pitman (Advanced Publishing Program), Boston, MA, 1985.7. P. Houston and T. P. Wihler, Second-order elliptic PDEs with discontinuous boundary data,IMA J. Numer. Anal. 32 (2012), no. 1, 48–74.8. J. M. Melenk, On generalized finite element methods, Ph.D. thesis, University of Maryland,1995.9. J. Nečas, Sur une méthode pour résoudre les équations aux dérivées partielles du type ellip-tique, voisine de la variationelle, Ann. Scuola Norm. Sup., Pisa 16 (1962), 305–326.10. J. Nitsche, Über ein Variationsprinzip zur Lösung von Dirichlet Problemen bei Verwendungvon Teilräumen, die keinen Randbedingungen unterworfen sind, Abh. Math. Sem. Univ. Ham-burg 36 (1971), 9–15.

A Nitsche Finite Element Approach for Elliptic Problems with Discontinuous Dirichlet Boundary Conditions

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