Abstract
The solutions of boundary value problems may contain singularities and/or have layers , where the solution changes rapidly. For such non-smooth functions, the application of pointwise interpolation is not well defined and in the presence of layers the use of regular and uniform meshes is not optimal in some sense. For these reasons quasi-interpolation operators for non-smooth functions over polytopal meshes are introduced and analysed in this chapter. In particular, operators of Clément- and Scott–Zhang-type are studied. Furthermore, the notion of anisotropic meshes is introduced. These meshes do not satisfy the classical regularity properties used in the approximation theory and thus they have to be treated in a special way. However, such meshes allow the accurate and efficient approximation of functions featuring anisotropic behaviours near boundary or interior layers.
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Agmon, S.: Lectures on Elliptic Boundary Value Problems. Prepared for publication by B. Frank Jones, Jr. with the assistance of George W. Batten, Jr. Van Nostrand Mathematical Studies, No. 2. D. Van Nostrand Co., Inc., Princeton (1965)
Antonietti, P.F., Beirão da Veiga, L., Verani, M.: A mimetic discretization of elliptic obstacle problems. Math. Comput. 82(283), 1379–1400 (2013)
Apel, T.: Anisotropic finite elements: local estimates and applications. In: Advances in Numerical Mathematics. B. G. Teubner, Stuttgart (1999)
Brenner, S.C., Scott, L.R.: The Mathematical Theory of Finite Element Methods. Texts in Applied Mathematics, vol. 15, 2nd edn. Springer, New York (2002)
Clément, P.: Approximation by finite element functions using local regularization. ESAIM Math. Model. Numer. Anal. 9(R-2), 77–84 (1975)
Dupont, T., Scott, R.: Polynomial approximation of functions in Sobolev spaces. Math. Comp. 34(150), 441–463 (1980)
Formaggia, L., Perotto, S.: New anisotropic a priori error estimates. Numer. Math. 89(4), 641–667 (2001)
Formaggia, L., Perotto, S.: Anisotropic error estimates for elliptic problems. Numer. Math. 94(1), 67–92 (2003)
Huang, W.: Mathematical principles of anisotropic mesh adaptation. Commun. Comput. Phys. 1(2), 276–310 (2006)
Huang, W., Kamenski, L., Lang, J.: A new anisotropic mesh adaptation method based upon hierarchical a posteriori error estimates. J. Comput. Phys. 229(6), 2179–2198 (2010)
Kunert, G.: An a posteriori residual error estimator for the finite element method on anisotropic tetrahedral meshes. Numer. Math. 86(3), 471–490 (2000)
Loseille, A., Alauzet, F.: Continuous mesh framework part I: well-posed continuous interpolation error. SIAM J. Numer. Anal. 49(1), 38–60 (2011)
Payne, L.E., Weinberger, H.F.: An optimal Poincaré inequality for convex domains. Arch. Ration. Mech. Anal. 5, 286–292 (1960)
Schneider, R.: A review of anisotropic refinement methods for triangular meshes in FEM. In: Apel, T., Steinbach, O. (eds.) Advanced Finite Element Methods and Applications, pp. 133–152. Springer, Berlin (2013)
Scott, L.R., Zhang, S.: Finite element interpolation of nonsmooth functions satisfying boundary conditions. Math. Comput. 54(190), 483–493 (1990)
Veeser, A., Verfürth, R.: Poincaré constants for finite element stars. IMA J. Numer. Anal. 32(1), 30–47 (2012)
Verfürth, R.: A Posteriori Error Estimation Techniques for Finite Element Methods. Numerical Mathematics and Scientific Computation. Oxford University Press, Oxford (2013)
Weißer, S.: Residual error estimate for BEM-based FEM on polygonal meshes. Numer. Math. 118(4), 765–788 (2011)
Weißer, S.: Residual based error estimate and quasi-interpolation on polygonal meshes for high order BEM-based FEM. Comput. Math. Appl. 73(2), 187–202 (2017)
Weißer, S.: Anisotropic polygonal and polyhedral discretizations in finite element analysis. ESAIM Math. Model. Numer. Anal. 53(2), 475–501 (2019)
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Weißer, S. (2019). Interpolation of Non-smooth Functions and Anisotropic Polytopal Meshes. In: BEM-based Finite Element Approaches on Polytopal Meshes. Lecture Notes in Computational Science and Engineering, vol 130. Springer, Cham. https://doi.org/10.1007/978-3-030-20961-2_3
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DOI: https://doi.org/10.1007/978-3-030-20961-2_3
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