Abstract
Sometimes boundary value problems have isolated regions where the solution changes rapidly. Therefore, when solving numerically, one needs a fine grid to capture the high activity. The fine grid can be implemented as a composite coarse-fine grid or as a global fine grid. One cheaper way of obtaining the composite grid solution is the use of the local defect correction technique. The technique is an algorithm that combines a global coarse grid solution and a local fine grid solution in an iterative way to estimate the solution on the corresponding composite grid. The algorithm is relatively new and its convergence properties have not been studied for the boundary element method. In this paper the objective is to determine convergence properties of the algorithm for the boundary element method. First, we formulate the algorithm as a fixed point iterative scheme, which has also not been done before for the boundary element method, and then study the properties of the iteration matrix. Results show that we can always expect convergence. Therefore, the algorithm opens up a real alternative for application in the boundary element method for problems with localised regions of high activity.
Highlights
Often boundary value problems have small localised regions of high activity where the solution varies rapidly compared to the rest of the domain
One way of approximating this composite grid solution that is simple and less complex than directly solving on the composite grid is by Local Defect Correction (LDC)
In Ferket et al [Ferket and Reusken (1996); Hackbusch (1984)] LDC has been shown to be a useful way of approximating the composite grid solution in which a global coarse grid solution is improved by a local fine grid solution, through a process whereby the right hand side of the global coarse grid problem system of equations is corrected by the defect of a local fine grid approximation
Summary
Often boundary value problems have small localised regions of high activity where the solution varies rapidly compared to the rest of the domain. In Ferket et al [Ferket and Reusken (1996); Hackbusch (1984)] LDC has been shown to be a useful way of approximating the composite grid solution in which a global coarse grid solution is improved by a local fine grid solution, through a process whereby the right hand side of the global coarse grid problem system of equations is corrected by the defect of a local fine grid approximation The properties for this method in FDM have been well studied, see for instance [Anthonissen (2001); Ferket and Reusken (1996); Hackbusch (1984); Minero, Anthonissen and Mattheij (2006)].
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