Low regularity estimates for CutFEM approximations of an elliptic problem with mixed boundary conditions

Burman, Erik; Hansbo, Peter; Larson, Mats

doi:10.1090/mcom/3875

Low regularity estimates for CutFEM approximations of an elliptic problem with mixed boundary conditions
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by Erik Burman, Peter Hansbo and Mats G. Larson

Math. Comp. 93 (2024), 35-54

DOI: https://doi.org/10.1090/mcom/3875

Published electronically: August 7, 2023

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Abstract:

We show error estimates for a cut finite element approximation of a second order elliptic problem with mixed boundary conditions. The error estimates are of low regularity type where we consider the case when the exact solution $u \in H^s$ with $s\in (1,3/2]$. For Nitsche type methods this case requires special handling of the terms involving the normal flux of the exact solution at the the boundary. For Dirichlet boundary conditions the estimates are optimal, whereas in the case of mixed Dirichlet-Neumann boundary conditions they are suboptimal by a logarithmic factor.

References

Erik Burman, Susanne Claus, Peter Hansbo, Mats G. Larson, and André Massing, CutFEM: discretizing geometry and partial differential equations, Internat. J. Numer. Methods Engrg. 104 (2015), no. 7, 472–501. MR 3416285, DOI 10.1002/nme.4823
Erik Burman and Peter Hansbo, Fictitious domain finite element methods using cut elements: II. A stabilized Nitsche method, Appl. Numer. Math. 62 (2012), no. 4, 328–341. MR 2899249, DOI 10.1016/j.apnum.2011.01.008
Erik Burman, Peter Hansbo, and Mats G. Larson, A cut finite element method with boundary value correction, Math. Comp. 87 (2018), no. 310, 633–657. MR 3739212, DOI 10.1090/mcom/3240
Erik Burman, Peter Hansbo, Mats G. Larson, and Sara Zahedi, Cut finite element methods for coupled bulk-surface problems, Numer. Math. 133 (2016), no. 2, 203–231. MR 3489084, DOI 10.1007/s00211-015-0744-3
W. Dahmen, B. Faermann, I. G. Graham, W. Hackbusch, and S. A. Sauter, Inverse inequalities on non-quasi-uniform meshes and application to the mortar element method, Math. Comp. 73 (2004), no. 247, 1107–1138. MR 2047080, DOI 10.1090/S0025-5718-03-01583-7
Daniele Antonio Di Pietro and Alexandre Ern, Mathematical aspects of discontinuous Galerkin methods, Mathématiques & Applications (Berlin) [Mathematics & Applications], vol. 69, Springer, Heidelberg, 2012. MR 2882148, DOI 10.1007/978-3-642-22980-0
Alexandre Ern and Jean-Luc Guermond, Analysis of the edge finite element approximation of the Maxwell equations with low regularity solutions, Comput. Math. Appl. 75 (2018), no. 3, 918–932. MR 3766493, DOI 10.1016/j.camwa.2017.10.017
Lawrence C. Evans, Partial differential equations, 2nd ed., Graduate Studies in Mathematics, vol. 19, American Mathematical Society, Providence, RI, 2010. MR 2597943, DOI 10.1090/gsm/019
Thirupathi Gudi, A new error analysis for discontinuous finite element methods for linear elliptic problems, Math. Comp. 79 (2010), no. 272, 2169–2189. MR 2684360, DOI 10.1090/S0025-5718-10-02360-4
Nora Lüthen, Mika Juntunen, and Rolf Stenberg, An improved a priori error analysis of Nitsche’s method for Robin boundary conditions, Numer. Math. 138 (2018), no. 4, 1011–1026. MR 3778343, DOI 10.1007/s00211-017-0927-1
Giuseppe Savaré, Regularity and perturbation results for mixed second order elliptic problems, Comm. Partial Differential Equations 22 (1997), no. 5-6, 869–899. MR 1452171, DOI 10.1080/03605309708821287