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Du, Qikui; Yu, Dehao

Dirichlet-Neumann alternating algorithm based on the natural boundary reduction for time-dependent problems over an unbounded domain. (English) Zbl 1013.65102

Appl. Numer. Math. 44, No. 4, 471-486 (2003).
Summary: A new iterative algorithm to solve a time-dependent problem over an unbounded domain is suggested. This method is based on the natural boundary reduction and is suitable for solving initial boundary value problems of time-dependent wave equation over an unbounded domain.
Firstly, an circular artificial boundary is introduced. Then the original unbounded domain is decomposed into a bounded domain and an exterior unbounded domain outside the artificial boundary. The natural integral equation obtained by the natural boundary reduction is used as a boundary condition on the artificial boundary.
Secondly, a Dirichlet-Neumann (D-N) alternating iterative algorithm is constructed. The algorithm is equivalent to preconditioned Richardson iteration method.
Thirdly, numerical studies are performed by finite element methods, and the results demonstrate the effectiveness of this algorithm. Finally, some remarks are presented.

Cited in 6 Documents

MSC:

65M20 Method of lines for initial value and initial-boundary value problems involving PDEs
65M55 Multigrid methods; domain decomposition for initial value and initial-boundary value problems involving PDEs
65M60 Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs
65F10 Iterative numerical methods for linear systems
65F35 Numerical computation of matrix norms, conditioning, scaling
35L05 Wave equation

Keywords:

time-dependent problem; natural boundary reduction (NBR); Dirichlet-Neumann alternating algorithm; semidiscretization; unbounded domain; domain decomposition; preconditioning; numerical examples; initial boundary value problem; wave equation; artificial boundary; Richardson iteration method; finite element method
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\textit{Q. Du} and \textit{D. Yu}, Appl. Numer. Math. 44, No. 4, 471--486 (2003; Zbl 1013.65102)
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