Discontinuous Galerkin finite element methods for a forward-backward heat equation

Department of Mathematical Sciences, University of Cincinnati, P.O. Box 210025, Cincinnati, OH 45221, USA
Applied Numerical Mathematics (Impact Factor: 1.22). 09/1998; 28(1):37-44. DOI: 10.1016/S0168-9274(98)00011-7


A space-time finite element method is introduced to solve a model forward-backward heat equation. The scheme uses the discontinuous Galerkin method for the time discretization. An error analysis for the scheme is presented.

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    • "Much work has been done for the one space dimensional case, including the investigation of a finite difference scheme[20] and methods of transforming the problem to a first order system[2], weighted least squares method[3], continuous and discontinuous Galerkin finite element methods[10] [11], and Galerkin and weighted Galerkin methods[17]. Domain decomposition algorithms and iterative methods are also developed for the solution of the problem[5] [6] [14]. "
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    ABSTRACT: A finite difference method is introduced to solve the forward-backward heat equation in two space dimensions. In this procedure, the backward and forward difference scheme in two subdomains and a coarse-mesh second-order central difference scheme at the middle interface are used. Maximum norm error estimate for the procedure is derived. Then an iterative method based on domain decomposition is presented for the numerical scheme and the convergence of the given method is established. Then numerical experiments are presented to support the theoretical analysis. KeywordsForward-backward heat equation-coarse mesh-iterative method MR Subject Classification65N20-65N10-35K05-65N55
    Applied Mathematics 03/2010; 25(1):101-111. DOI:10.1007/s11766-010-1812-1 · 0.34 Impact Factor
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    • "Aziz and Liu [16] [17] have transformed the backward–forward diffusion equation into a first-order system of symmetric-positive differential equations in the sense of Friedrichs which they solve using finite elements. In contrast, Lu [18] and French [19] [20] have developed Galerkin methods without transforming the problem. Ref. [21] provides an overview of these finite element methods. "
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    Journal of Quantitative Spectroscopy and Radiative Transfer 03/2008; 109(5):727-740. DOI:10.1016/j.jqsrt.2007.09.011 · 2.65 Impact Factor
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    • "Aziz and Liu [3] transformed the second order (1) into a first order system of symmetric positive definite type and solved it using a weighted least-squares method. The finite element of the discontinuous Galerkin method for the time discretization has been considered in [6]. "
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    ABSTRACT: In this article we analyzed the convergence of the Schwarz waveform relaxation method for solving the forward-backward heat equation. Numerical results are presented for a specific type of model problem.
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