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

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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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References

  1. 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)

    Article  MathSciNet  MATH  Google Scholar 

  2. 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)

    Article  MathSciNet  MATH  Google Scholar 

  3. 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)

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

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

    MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

  8. 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)

    Article  MathSciNet  MATH  Google Scholar 

  9. 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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  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  Google Scholar 

  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  Google Scholar 

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

    MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

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

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Acknowledgments

We would like to thank the anonymous referee for the valuable suggestions. The research of B. Li was partially supported by the start-up grant of The Hong Kong Polytechnic University, and was partially carried during a research stay at University of Tübingen, funded by the Alexandre von Humboldt foundation and NSFC 11301262.

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Authors and Affiliations

  1. Department of Mathematics, University of Connecticut, Storrs, USA

    Dmitriy Leykekhman

  2. Department of Applied Mathematics, The Hong Kong Polytechnic University, Kowloon, Hong Kong

    Buyang Li

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  1. Dmitriy Leykekhman
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  2. Buyang Li
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Correspondence to Buyang Li.

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Leykekhman, D., Li, B. Maximum-norm stability of the finite element Ritz projection under mixed boundary conditions. Calcolo 54, 541–565 (2017). https://doi.org/10.1007/s10092-016-0198-8

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