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Implementation of an operator-splitting finite element method for high-dimensional parabolic problems

by Ganesan, S.; Tobiska, L..

Series: 2011-04, Preprints

65M60 Finite elements, ~Rayleigh-Ritz and Galerkin methods, finite methods
65Y20 Complexity and performance of numerical algorithms
35K20 Initial-boundary value problems for second-order parabolic equations

The operator-splitting method in the context of the finite element method for solving high-dimensional parabolic equations is presented. The key idea is to split the high-dimensional equation into a system of low-dimensional equations, and solve each equation in the system separately. The equivalence up to a perturbation term of order $(\delta t)^2$ between the high-dimensional problem and the operator-split system of low-dimensional subproblems is shown for the backward Euler time discretization scheme. Furthermore, two variants of fully-practical algorithms, (i) quadrature point based algorithm, and (ii) nodal point based algorithm, in the context of the finite element method are presented. Both the quadrature and nodal point based operator-splitting algorithms are validated using a three-dimensional (3D) test problem. The numerical results obtained with the full 3D computations and the operator-split 2D+1D computations are found to be in a good agreement with the analytical solution. Further, the optimal order of convergence is obtained in both variants of the operator-splitting algorithms.

Operator-splitting method, Finite element method, Parabolic equations, High-dimensional problems