Two‐grid methods for semilinear interface problems |
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| Authors: | Michael Holst Ryan Szypowski Yunrong Zhu |
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| Institution: | 1. Department of Mathematics, University of California San Diego, La Jolla, California 92093;2. Department of Mathematics and Statistics, California State Polytechnic University, Pomona, Pomona, California 91768;3. Department of Mathematics, Idaho State University, Pocatello, Idaho 83209 |
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| Abstract: | In this article, we consider two‐grid finite element methods for solving semilinear interface problems in d space dimensions, for d = 2 or d = 3. We consider semilinear problems with discontinuous diffusion coefficients, which includes problems containing subcritical, critical, and supercritical nonlinearities. We establish basic quasioptimal a priori error estimates for Galerkin approximations. We then design a two‐grid algorithm consisting of a coarse grid solver for the original nonlinear problem, and a fine grid solver for a linearized problem. We analyze the quality of approximations generated by the algorithm and show that the coarse grid may be taken to have much larger elements than the fine grid, and yet one can still obtain approximation quality that is asymptotically as good as solving the original nonlinear problem on the fine mesh. © 2013 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2013 |
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| Keywords: | a priori L∞ estimates discrete a priori L∞ estimates Galerkin methods interface problems Poisson– Boltzmann equation quasioptimal a priori error estimates semilinear partial differential equations two‐grid methods |
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