×

Preconditioning the biharmonic equation by multilevel iterations. (English) Zbl 1034.65507

Summary: A preconditioned conjugate gradient iterative method is studied for the numerical solution of the biharmonic equation with Dirichlet boundary conditions for the solution and its normal derivative in a rectangular domain \(\Omega\). We consider two discretizations of the differential equation: by standard finite differences and by quadratic splines from \(S_2 \subset C^1 (\Omega)\). The proposed preconditioner is constructed by first approximating the biharmonic operator by changing the boundary conditions thus allowing factorizing the new operator into product of two Laplacians. This product is then preconditioned by a recently proposed algebraic multilevel technique based on corresponding discretizations of the Laplace operator. The resulting multilevel preconditioner is used in the preconditioned conjugate gradient method giving rise to a multilevel algorithm with a total cost of \(O(N^{5\over 4}\log {1\over\varepsilon})\), where \(N\) is the number of the unknowns and \(\varepsilon\) is the accuracy in the conjugate gradient method. Various numerical tests illustrating the properties of the algorithm obtained are presented. In particular, an \(O(h^{-1})\) relative condition number of the constructed multilevel preconditioner with respect to the corresponding matrices was observed in the tests confirming the theoretical estimate.

MSC:

65N06 Finite difference methods for boundary value problems involving PDEs
65N55 Multigrid methods; domain decomposition for boundary value problems involving PDEs
65F10 Iterative numerical methods for linear systems
65F35 Numerical computation of matrix norms, conditioning, scaling
35J40 Boundary value problems for higher-order elliptic equations
PDFBibTeX XMLCite