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01.06.2017

Maximum-norm stability of the finite element Ritz projection under mixed boundary conditions

verfasst von: Dmitriy Leykekhman, Buyang Li

Erschienen in: Calcolo | Ausgabe 2/2017

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Abstract

As a model of the second order elliptic equation with non-trivial boundary conditions, we consider the Laplace equation with mixed Dirichlet and Neumann boundary conditions on convex polygonal domains. Our goal is to establish that finite element discrete harmonic functions with mixed Dirichlet and Neumann boundary conditions satisfy a weak (Agmon–Miranda) discrete maximum principle, and then prove the stability of the Ritz projection with mixed boundary conditions in \(L^\infty \) norm. Such results have a number of applications, but are not available in the literature. Our proof of the maximum-norm stability of the Ritz projection is based on converting the mixed boundary value problem to a pure Neumann problem, which is of independent interest.

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Alkhutov, Y., Maz’ya, V.G.: \({L}^{1, p}\)-coercitivity and estimates of the Green function of the Neumann problem in a convex domain. J. Math. Sci. 196(3), 245–261 (2014)MathSciNetCrossRefMATH

Apel, T., Pfefferer, J., Rösch, A.: Finite element error estimates for Neumann boundary control problems on graded meshes. Comput. Optim. Appl. 52(1), 3–28 (2012)MathSciNetCrossRefMATH

Bakaev, N.Y., Thomée, V., Wahlbin, L.B.: Maximum-norm estimates for resolvents of elliptic finite element operators. Math. Comput. 72(244), 1597–1610 (2003) (electronic)

Bernardi, C., Dauge, M., Maday, Y.: Polynomials in the Sobolev World. Working paper or preprint, September 2007

Ciarlet, P.G.: The Finite Element Method for Elliptic Problems. North-Holland, Amsterdam (1978)MATH

Crouzeix, M., Larsson, S., Thomée, V.: Resolvent estimates for elliptic finite element operators in one dimension. Math. Comput. 63(207), 121–140 (1994)MathSciNetCrossRefMATH

Dauge, M.: Elliptic Boundary Value Problems on Corner Domains: Smoothness and Asymptotics of Solutions. Lecture Notes in Mathematics, vol. 1341. Springer, Berlin (1988)

Demlow, A., Leykekhman, D., Schatz, A.H., Wahlbin, L.B.: Best approximation property in the \(W^{1}_{\infty }\) norm for finite element methods on graded meshes. Math. Comput. 81(278), 743–764 (2012)MathSciNetCrossRefMATH

Fabrikant, V.I.: Mixed Boundary Value Problems of Potential Theory and Their Applications in Engineering. Mathematics and its Applications, vol. 68. Kluwer Academic Publishers Group, Dordrecht (1991)

10.

Fried, I.: On the optimality of the pointwise accuracy of the finite element solution. Int. J. Numer. Methods Eng. 15(3), 451–456 (1980)MathSciNetCrossRefMATH

11.

Geissert, M.: Discrete maximal \(L_p\) regularity for finite element operators. SIAM J. Numer. Anal. 44(2), 677–698 (2006) (electronic)

12.

Geissert, M.: Applications of discrete maximal \(L_p\) regularity for finite element operators. Numer. Math. 108(1), 121–149 (2007)MathSciNetCrossRefMATH

13.

Gilbarg, D.: Trudinger, N.S.: Elliptic Partial Differential Equations of Second Order, 2nd edn. Classics in Mathematics. Springer, Berlin (2001)

14.

Grisvard, P.: Elliptic Problems in Nonsmooth Domains. Monographs and Studies in Mathematics, vol. 24. Pitman (Advanced Publishing Program), Boston (1985)

15.

Guzmán, J., Leykekhman, D.: Pointwise error estimates of finite element approximations to the Stokes problem on convex polyhedra. Math. Comput. 81(280), 1879–1902 (2012)MathSciNetCrossRefMATH

16.

Guzmán, J., Leykekhman, D., Rossmann, J., Schatz, A.H.: Hölder estimates for Green’s functions on convex polyhedral domains and their applications to finite element methods. Numer. Math. 112(2), 221–243 (2009)MathSciNetCrossRefMATH

17.

Haverkamp, R.: Eine Aussage zur \(L_{\infty }\)-Stabilität und zur genauen Konvergenzordnung der \(H^{1}_{0}\)-Projektionen. Numer. Math. 44(3), 393–405 (1984)MathSciNetCrossRefMATH

18.

Leykekhman, D., Vexler, B.: Finite element pointwise results on convex polyhedral domains. SIAM J. Numer. Anal. 54(2), 561–587 (2016)MathSciNetCrossRefMATH

19.

Leykekhman, D., Vexler, B.: Pointwise best approximation results for Galerkin finite element solutions of parabolic problems. SIAM J. Numer. Anal. 54(3), 1365–1384 (2016)

20.

Leykekhman, D., Vexler, B.: A priori error estimates for threedimensional parabolic optimal control problems with pointwise control. SIAM J. Control. Optim. (2016)

21.

Li, B.: Maximum-norm stability and maximal \(L^p\) regularity of FEMs for parabolic equations with Lipschitz continuous coefficients. Numer. Math. 131(3), 489–516 (2015)MathSciNetCrossRefMATH

22.

Li, B., Sun, W.: Regularity of the diffusion-dispersion tensor and error analysis of Galerkin FEMs for a porous medium flow. SIAM J. Numer. Anal. 53(3), 1418–1437 (2015)MathSciNetCrossRefMATH

23.

Li, B., Sun, W.: Maximal \(L^p\) analysis of finite element solutions for parabolic equations with nonsmooth coefficients in convex polyhedra. Math. Comput. (in press). arXiv:1501.07345

24.

Nitsche, J.: \(L_{\infty }\)-convergence of finite element approximations. In: Mathematical Aspects of Finite Element Methods (Proc. Conf., Consiglio Naz. delle Ricerche (C.N.R.), Rome, 1975), pp. 261–274. Lecture Notes in Mathematics, vol. 606. Springer, Berlin (1977)

25.

Nitsche, J.A., Schatz, A.H.: Interior estimates for Ritz–Galerkin methods. Math. Comput. 28, 937–958 (1974)MathSciNetCrossRefMATH

26.

Pernavs, N.: On a mixed boundary value problem of harmonic functions. Proc. Am. Math. Soc. 7, 127–130 (1956)MathSciNetCrossRefMATH

27.

Rannacher, R., Scott, R.: Some optimal error estimates for piecewise linear finite element approximations. Math. Comput. 38(158), 437–445 (1982)MathSciNetCrossRefMATH

28.

Schatz, A.H., Wahlbin, L.B.: Interior maximum norm estimates for finite element methods. Math. Comput. 31(138), 414–442 (1977)MathSciNetCrossRefMATH

29.

Schatz, A.H., Wahlbin, L.B.: On the quasi-optimality in \(L_{\infty }\) of the \(\dot{H}^{1}\)-projection into finite element spaces. Math. Comput. 38(157), 1–22 (1982)MATH

30.

Schatz, A.H.: A weak discrete maximum principle and stability of the finite element method in \(L_{\infty }\) on plane polygonal domains. I. Math. Comput. 34(149), 77–91 (1980)MATH

31.

Strang, G., Fix, G.J.: An Analysis of the Finite Element Method. Wellesley-Cambridge Press, Wellesley (1988)MATH

32.

Thomée, V., Wahlbin, L.B.: Stability and analyticity in maximum-norm for simplicial Lagrange finite element semidiscretizations of parabolic equations with Dirichlet boundary conditions. Numer. Math. 87(2), 373–389 (2000)MathSciNetCrossRefMATH

33.

Thomée, V.: Galerkin Finite Element Methods for Parabolic Problems. Springer Series in Computational Mathematics, vol. 25, 2nd edn. Springer, Berlin (2006)

Titel: Maximum-norm stability of the finite element Ritz projection under mixed boundary conditions
verfasst von: Dmitriy Leykekhman
Buyang Li
Publikationsdatum: 01.06.2017
Verlag: Springer Milan
Erschienen in: Calcolo / Ausgabe 2/2017
Print ISSN: 0008-0624
Elektronische ISSN: 1126-5434
DOI: https://doi.org/10.1007/s10092-016-0198-8

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Maximum-norm stability of the finite element Ritz projection under mixed boundary conditions

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