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In this paper we propose and analyze finite element discontinuous Galerkin methods for the one- and two-dimensional stochastic Maxwell equations with multiplicative noise. The discrete energy law of the semi-discrete DG methods were studied. Optimal error estimate of the semi-discrete method is obtained for the one-dimensional case, and the two-dimensional case on both rectangular meshes and triangular meshes under certain mesh assumptions. Strong Taylor 2.0 scheme is used as the temporal discretization. Both one- and two-dimensional numerical results are presented to validate the theoretical analysis results.
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Efficient Exponential Integrator Finite Element Method for Semilinear Parabolic Equations
In this paper, we propose an efficient exponential integrator finite element method for solving a class of semilinear parabolic equations in rectangular domains. The proposed method first performs the spatial discretization of the model equation using the nite element approximation with continuous multilinear rectangular basis functions, and then takes the explicit exponential Runge-Kutta approach for time integration of the resulting semi-discrete system to produce fully-discrete numerical solution. Under certain regularity assumptions, error estimates measured in H1-norm are successfully derived for the proposed schemes with one and two RK stages. More remarkably, the mass and coefficient matrices of the proposed method can be simultaneously diagonalized with an orthogonal matrix, which provides a fast solution process based on tensor product spectral decomposition and fast Fourier transform. Various numerical experiments in two and three dimensions are also carried out to validate the theoretical results and demonstrate the excellent performance of the proposed method.
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- Award ID(s):
- 2109633
- PAR ID:
- 10544600
- Publisher / Repository:
- SIAM
- Date Published:
- Journal Name:
- SIAM Journal on Scientific Computing
- Volume:
- 45
- Issue:
- 4
- ISSN:
- 1064-8275
- Page Range / eLocation ID:
- A1545-A1570
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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- Free Publicly Accessible Full Text
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Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.1137/22M1498127
