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Condition Number of the Boundary Element Method Matrices
W. Dijkstra
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Project description:
The Boundary Element Method (BEM) is often used to solve Boundary Value Problems (BVP). It transforms the
BVP into an integral equation and after discretisation of the boundary into a set of linear algebraic
equations. The system matrix that appears in the set of algebraic equations is dense. As a consequence, the
condition number of the matrix is large, which leads to a low stability with respect to perturbations.
It is a well-known fact that the condition number of the system matrix is at least order N, where N
is the number of boundary elements. In literature, the Dirichlet BVP
for the Laplace equation is studied for two domains: a circle and an ellips. In both
cases analytical expressions for the condition number are derived. The results show how the condition number
depends on the radius of the circle or the aspect ratio of the ellips. However, all papers deal with Dirichlet
conditions on the boundary of the circle. We will extend the research by studying mixed boundary conditions.
In this project we investigate the condition number of the BEM-matrix corresponding to the Laplace equation
on a circle with mixed boundary conditions. We estimate the condition number for this situation and study
the effect of the number of elements N and the radius R of the circle onto this condition number.
Also, we investigate the role of m, the number of elements that have Dirichlet conditions.
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