The Iterative Solver RISOLV with Application to the Exterior Helmholtz Problem

• Authors: Hillel Tal-Ezer;Eli Turkel
• Venue: SIAM Journal on Scientific Computing
• Year: 2010

The innermost computational kernel of many large-scale scientific applications is often a large set of linear equations of the form $Ax=b$ which typically consumes a significant portion of the overall computational time required by the simulation. The traditional approach for solving this problem is to use direct methods. This approach is often preferred in industry because direct solvers are robust and effective for moderate size problems. However, direct methods can consume a huge amount of memory, and CPU time, in large-scale cases. In these cases, iterative techniques are the only viable alternative. Unfortunately, iterative methods lack the robustness of direct methods. The situation is especially difficult when the matrix is nonsymmetric. A lot of research has been devoted to trying to develop a robust iterative algorithm for nonsymmetric systems. The present paper describes a new robust and efficient algorithm aimed at solving iteratively nonsymmetric linear systems. It is based on looking for an approximation to the “optimal” polynomial $P_m(z)$ which satisfies $||P_m(z)||_{\infty}=\min_{Q\in\Pi_m}||Q(z)||_{\infty}$, $z\in D$, where $\Pi_m$ is the set of all polynomials of degree $m$ which satisfies $Q_m(0)=1$ and $D$ is a domain in the complex plane which includes all the eigenvalues of $A$. The resulting algorithm is an efficient one, especially in the case where we have a set of linear systems which share the same matrix $A$. We present several applications, including the exterior Helmholtz problem, which leads to a large indefinite, nonsymmetric, and complex system.