Fully discrete A‐ϕ finite element method for Maxwell's equations with nonlinear conductivity |
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| Authors: | Tong Kang Tao Chen Huai Zhang Kwang Ik Kim |
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| Institution: | 1. Department of Applied Mathematics, School of Sciences, Communication University of China, , Beijing, 100024 People's Republic of China;2. Key Laboratory of Computational Geodynamics, University of Chinese Academy of Sciences, , Beijing, 100049 People's Republic of China;3. Department of Mathematics, Pohang University of Science and Technology, , Pohang, 790‐784 Republic of Korea |
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| Abstract: | This article is devoted to the study of a fully discrete A ‐ finite element method to solve nonlinear Maxwell's equations based on backward Euler discretization in time and nodal finite elements in space. The nonlinearity is owing to a field‐dependent conductivity with the power‐law form  . We design a nonlinear time‐discrete scheme for approximation in suitable function spaces. We show the well‐posedness of the problem, prove the convergence of the semidiscrete scheme based on the boundedness of the second derivative in the dual space and derive its error estimate. The Minty–Browder technique is introduced to obtain the convergence of the nonlinear term. Finally, we discuss the error estimate for the fully discretized problem and support the theoretical result by two numerical experiments. © 2014 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 30: 2083–2108, 2014 |
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| Keywords: | A‐ϕ method convergence error estimates nodal elements nonlinear Maxwell's equations |
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