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 and methods of transforming the problem to a first order system, weighted least squares method, continuous and discontinuous Galerkin finite element methods , and Galerkin and weighted Galerkin methods. Domain decomposition algorithms and iterative methods are also developed for the solution of the problem  . "
[Show abstract][Hide abstract] 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
Preview · Article · Mar 2010 · Applied Mathematics
"Aziz and Liu   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  and French   have developed Galerkin methods without transforming the problem. Ref.  provides an overview of these finite element methods. "
[Show abstract][Hide abstract] ABSTRACT: We study the numerical solution of the Fokker–Planck equation. This equation gives a good approximation to the radiative transport equation when scattering is peaked sharply in the forward direction which is the case for light propagation in tissues, for example. We derive first the numerical solution for the problem with constant coefficients. This numerical solution is constructed as an expansion in plane wave solutions. Then we extend that result to take into account coefficients that vary spatially. This extension leads to a coupled system of initial and final value problems. We solve this system iteratively. Numerical results show the utility of this method.
Preview · Article · Mar 2008 · Journal of Quantitative Spectroscopy and Radiative Transfer
"Aziz and Liu  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 . "
[Show abstract][Hide abstract] 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.
Preview · Article · Nov 2007 · Journal of Computational and Applied Mathematics