Fast parallel solvers for fourth-order boundary value problems

Abstract

Publisher Summary This chapter discusses the fast parallel solvers for fourth-order boundary value problems. The first biharmonic boundary value problem in mixed weak formulation is considered. The finite element discretization of this problem leads to a system of linear algebraic equations with a symmetric indefinite matrix. The chapter discusses the three possibilities for solving this system of equations efficiently— namely, the preconditioned conjugate gradient method for a corresponding Schur complement system, a conjugate gradient method of Bramble-Pasciak type and a multigrid method. Furthermore, the implementation of these solvers on a parallel computer with MIMD architecture is described. The numerical experiments presented show that these solution methods can be parallelized very efficiently. The number of iterations of the Schur complement pcg method and of the cg method of Bramble-Pasciak type grows with a factor of about √2, which confirms the theoretical result given in the chapter. The number of iterations of the multigrid method is nearly constant. If one uses the parallelization strategy proposed and if the problem size is large enough, then all methods have a very good parallel efficiency. For large problems, the multigrid algorithm is the fastest solver.