Projected Schur complement method for solving non‐symmetric systems arising from a smooth fictitious domain approach

Abstract

SUMMARYThis paper deals with a fast method for solving large-scale algebraic saddle-point systems arising from fictitious domain formulations of elliptic boundary value problems. A new variant of the fictitious domain approach is analyzed. Boundary conditions are enforced by control variables introduced on an auxiliary boundary located outside the original domain. This approach has a significantly higher convergence rate; however, the algebraic systems resulting from finite element discretizations are typically non-symmetric. The presented method is based on the Schur complement reduction. If the stiffness matrix is singular, the reduced system can be formulated again as another saddle-point problem. Its modification by orthogonal projectors leads to an equation that can be efficiently solved using a projected Krylov subspace method for non-symmetric operators. For this purpose, the projected variant of the BiCGSTAB algorithm is derived from the non-projected one. The behavior of the method is illustrated by examples, in which the BiCGSTAB iterations are accelerated by a multigrid strategy.

Preconditioners for regularized saddle point problems with an application for heterogeneous Darcy flow problems

“…Remark 3.9 An interesting problem where the off-diagonal block matrices C, B are different is presented in [25]. Here the saddle point structure arises from a fictitious domain approach to solve elliptic boundary value problems, where the boundary conditions on the auxiliary boundary outside the given domain are enforced by certain control variables.…”

A smooth fictitious domain/multiresolution method for elliptic equations on general domains

“…It is well known that the smoothness degree of the solution of a boundary-value problem does affect the order of convergence of a numerical solution [13]. J.Haslinger et al [7] investigated a new formulation of fictitious domain methods connecting in putting the Lagrange multiplier λ on a control boundary Γ located outside ofω to enforce the boundary condition on γ(see in Figure 1). We will call it the smooth fictitious domain/Lagrange multiplier method (SFDLM).…”

“…By shifting the Lagrange multipliers on a control boundary Γ located away from γ in the outer normal direction(as proposed in [7]), this new approach is shown to preserve the good approximation properties of multiresolution methods and to improve the accuracy order of the numerical solution on ω.…”

Parallel solution of contact shape optimization problems with Coulomb friction based on domain decomposition

“…We can use Total FETI method for the searching of solution to the limiting adjoint generalized equation. Our approach is based on [5].…”