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Published in:
2017 | OriginalPaper | Chapter
On the Definition of Dirichlet and Neumann Conditions for the Biharmonic Equation and Its Impact on Associated Schwarz Methods
Authors : Martin J. Gander, Yongxiang Liu
Published in: Domain Decomposition Methods in Science and Engineering XXIII
Publisher: Springer International Publishing
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Abstract
The biharmonic equation is a fourth order equation, and thus needs two boundary conditions, and not just one like Laplace’s equation. While the clamped boundary condition can be taken naturally as the “Dirichlet” condition, there are also other choices, and we are interested in studying the dependence of Schwarz methods on the choice one makes for the “Dirichlet” condition when defining the Schwarz algorithm. We show that the classical choice actually leads to a very slow Schwarz algorithm and makes the biharmonic equation appear to be difficult to solve by Schwarz methods. A different choice for the “Dirichlet” condition leads to a much faster Schwarz algorithm comparable to the Schwarz algorithm applied to the Laplace equation. We then show that optimized Schwarz methods can automatically correct this problem, independently of what choice one makes for the “Dirichlet” condition, and they are in all cases largely superior to the classical Schwarz method.