Abstract
A model second-order elliptic equation on a general convex polyhedral domain in three dimensions is considered. The aim of this paper is twofold: First sharp Hölder estimates for the corresponding Green’s function are obtained. As an applications of these estimates to finite element methods, we show the best approximation property of the error in \({W^1_{\infty}}\) . In contrast to previously known results, \({W_p^{2}}\) regularity for p > 3, which does not hold for general convex polyhedral domains, is not required. Furthermore, the new Green’s function estimates allow us to obtain localized error estimates at a point.
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References
Asadzadeh, M., Schatz, A.H., Wendland, W.: Asymptotic Error Expansions for the Finite Element Method for Second Order Elliptic Problems in \({\mathbb{R}^N}\) , N ≥ 2. I: Local Interior Expansions. Math. Comp. (2009, in press)
Asadzadeh, M., Schatz, A.H., Wendland, W.: A Non-Standard Approach to Richardson Extrapolation in the Finite Element Method for Second Order Elliptic Problems, preprint, Chalmers University of Technology, Göteborg University
Brenner, S., Scott, R.: Mathematical Theory of Finite Element Methods, 3rd edn. Texts in Applied Mathematics, vol. 15. Springer, New York (2008)
Carey, V.: A Posteriori Error Estimation via Recovered Gradients. Dissertation, Cornell University (2005)
Chen Z., Chen H.: Pointwise error estimates of discontinuous Galerkin methods with penalty for second-order elliptic problems. SIAM J. Numer. Anal. 42, 1146–1166 (2004)
Demlow A.: Localized pointwise a posteriori error estimates for gradients of piecewise linear finite element approximations to second-order quasilinear elliptic problems. SIAM J. Numer. Anal. 44(2), 494–514 (2006)
Demlow, A.: Weighted Residual Estimators for a Posteriori Estimation of Pointwise Gradient Errors in Quasilinear Elliptic Problems, preprint
Demlow A.: Sharply localized pointwise and \({W_\infty^{-1}}\) estimates for finite element methods for quasilinear problems. Math. Comp. 76, 1725–1741 (2007)
Douglas J. Jr, Dupont T.: A Galerkin methods for a nonlinear Dirichlet problem. Math. Comp. 29, 689–696 (1975)
Dupont T., Scott R.: Polynomial approximation of functions in Sobolev spaces. Math. Comp. 34, 441–463 (1980)
Frehse J., Rannacher R.: Asymptotic L ∞-error estimates for linear finite element approximations of quasilinear boundary value problems. SIAM J. Numer. Anal. 15(2), 418–431 (1978)
Fromm S.: Potential space estimates for Green potentials in convex domains. Proc. Amer. Math. Soc. 119(1), 225–233 (1993)
Girault V., Nochetto R.H., Scott R.: Maximum-norm stability of the finite element Stokes projection. J. Math. Pures Appl. 84(9), 279–330 (2005)
Guzmán J.: Pointwise error estimates for discontinuous Galerkin methods with lifting operators for elliptic problems. Math. Comp. 75, 1067–1085 (2006)
Grüter M., Widman K.-O.: The Green function for uniformly elliptic equations. Manuscr. Math. 37, 303–342 (1982)
Hoffmann W., Schatz A.H., Wahlbin L.B., Wittum G.: Asymptotically exact a posteriori estimators for the pointwise gradient error on each element in irregular meshes. Part 1: a smooth problem and globally quasi-uniform meshes. Math. Comp. 70, 897–909 (2001)
Kozlov V.A., Maz’ya V.G., Rossmann J.: Spectral Problems Assosoated with Corner Singularities of Solutions to Elliptic Equations, Mathematical Surveys and Monographs, vol. 85. Amer. Math. Soc., Providence (2001)
Krasovskii, J.P.: Properties of Green’s Function and Generalized Solutions of Elliptic Boundary Value Problems, Soviet Mathematics (Translations of Doklady Academy of Sciences of the USSR), 102, pp. 54–120 (1969)
Maz’ya V.G., Roßmann J.: On the Agmon-Miranda maximum principle for solutions of elliptic equations in polyhedral and polygonal domains. Ann. Glob. Anal. Geom. 9(3), 253–303 (1991)
Maz’ya V.G., Roßmann J.: Point estimates for Green’s matrix to boundary value problems for second order systems in a polyhedral cone. Z. Angew. Math. Mech. 82(5), 291–316 (2002)
Maz’ya V.G., Roßmann J.: Weighted L p estimates of solutions to boundary value problems for second order elliptic systems in polyhedral domains. Z. Angew. Math. Mech. 83(7), 435–467 (2003)
Maz’ya V.G., Roßmann J.: Schauder estimates for solutions to a mixed boundary value problem for the Stokes system in polyhedral domains. Math. Methods Appl. Sci. 29, 965–1017 (2006)
Natterer F.: Uber die punktweise Konvergenz finiter Elemente. Numer. Math. 25, 67–77 (1975)
Nitsche, J.A.: L ∞ convergence of finite element approximations. In: Proceedings Second Conference on Finite Elements, Rennes, France (1975)
Nitsche, J.A.: L ∞ Convergence of Finite Element Approximations, Mathematical Aspects of Finite Element Methods. Lecture Notes in Math., vol. 606, pp. 261–274. Springer, Berlin (1977)
Nitsche J.A., Schatz A.H.: Interior estimates for Ritz-Galerkin methods. Math. Comp. 28, 937–958 (1974)
Rannacher R.: Zur L ∞-Konvergenz linearer finiter Elemente beim Dirichlet-Problem. Math. Zeitschrift 149, 69–77 (1976)
Rannacher R., Scott R.: Some optimal error estimates for piecewise linear finite element approximations. Math. Comp. 158, 437–445 (1982)
Saavedra P., Scott R.: Variational formulation of a model free-boundary problem. Math. Comp. 57, 451–475 (1991)
Schatz A.H.: A weak discrete maximum principle and stability of the finite element method in L ∞ on the plane polygonal domains. I. Math. Comp. 34, 77–91 (1980)
Schatz A.H.: Pointwise error estimates and asymptotic error expansion inequalities for the finite element method on irregular grids: Part 1. Math. Comp. 67, 877–899 (1998)
Schatz A.H.: Pointwise error estimates and asymptotic error expansion inequalities for the finite element method on irregular grids. II. Interior estimates. SIAM J. Numer. Anal. 38(4), 1269–1293 (2000)
Schatz A.H.: Perturbations of forms and error estimates for the finite element method at a point, with an application to improved superconvergence error estimates for subspaces that are symmetric with respect to a point. SIAM J. Numer. Anal. 42(6), 2342–2365 (2005)
Schatz A.H., Wahlbin L.B.: Interior maximum norm estimates for finite element methods. Math. Comp. 31, 414–442 (1977)
Schatz A.H., Wahlbin L.B.: On the quasi-optimality in L ∞ of the \({\overset\circ{H^1}}\) -projection into finite element spaces. Math. Comp. 38, 1–22 (1982)
Schatz A.H., Wahlbin L.B.: Interior maximum norm estimates for finite element methods II. Math. Comp. 64, 907–928 (1995)
Scholz R.: A mixed method for 4th order problems using linear finite elements. RAIRO Anal. Numer. 12(1), 85–90 (1978)
Scott L.R.: Optimal L ∞ estimates for the finite element method. Math. Comp. 30, 681–697 (1976)
Wahlbin L.B.: Local behavior in finite element methods, Handbook of Numerical Analysis, vol, II. In: Ciarlet, P.G., Lions, J.L. (eds) Finite Element Methods (Part 1), pp. 355–522. Elsevier, Amsterdam (1991)
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J. Guzmán was supported by NSF grand DMS-0503050, D. Leykekhman was supported in part by NSF grands DMS-0240058 and DMS-0811167, and A. H. Schatz by NSF grand DMS-0612599.
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Guzmán, J., Leykekhman, D., Rossmann, J. et al. Hölder estimates for Green’s functions on convex polyhedral domains and their applications to finite element methods. Numer. Math. 112, 221–243 (2009). https://doi.org/10.1007/s00211-009-0213-y
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DOI: https://doi.org/10.1007/s00211-009-0213-y