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Discretization of the P oisson equation with non‐smooth data and emphasis on non‐convex domains
Author(s) -
Apel Thomas,
Nicaise Serge,
Pfefferer Johannes
Publication year - 2016
Publication title -
numerical methods for partial differential equations
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.901
H-Index - 61
eISSN - 1098-2426
pISSN - 0749-159X
DOI - 10.1002/num.22057
Subject(s) - discretization , mathematics , discretization error , regularization (linguistics) , dirichlet distribution , poisson's equation , mathematical analysis , dirichlet problem , poisson distribution , dirichlet boundary condition , partial differential equation , boundary value problem , computer science , artificial intelligence , statistics
Several approaches are discussed how to understand the solution of the Dirichlet problem for the Poisson equation when the Dirichlet data are non‐smooth such as if they are inL 2only. For the method of transposition (sometimes called very weak formulation ) three spaces for the test functions are considered, and a regularity result is proved. An approach of Berggren is recovered as the method of transposition with the second variant of test functions. A further concept is the regularization of the boundary data combined with the weak solution of the regularized problem. The effect of the regularization error is studied. The regularization approach is the simplest to discretize. The discretization error is estimated for a sequence of quasi‐uniform meshes. Since this approach turns out to be equivalent to Berggren's discretization his error estimates are rendered more precisely. Numerical tests show that the error estimates are sharp, in particular that the order becomes arbitrarily small when the maximal interior angle of the domain tends to 2 π .© 2016 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 32: 1433–1454, 2016