A fast interface solver for the biharmonic Dirichlet problem on polygonal domains
The authors describe a numerical method to solve the equation \[ \Delta^2 v=0 \text{ in } \Omega, \quad v|_\Gamma =\varphi |_\Gamma,\;\partial_n v|_\Gamma= \partial_n \varphi |_\Gamma, \] where \(\Omega\) is a convex polygonal domain with boundary \(\Gamma\) and \(\varphi\) is a sufficiently smooth function on a neighborhood of \(\Gamma\). A fast interface solver is defined with a complexity that is nearly optimal, i.e. up to a logarithmic factor it is linearly proportional with the number of degrees of freedom on the boundary \(\Gamma\). This is obtained by a clever reduction to a harmonic Dirichlet problem and another equation which can be handled by efficient matrix techniques. Both a careful analysis and a numerical example on a rectangular domain are presented.
scientific article; zbMATH DE number 1111590