Parallel preconditioning techniques for numerical solution of three dimensional partial differential equations

Sivas, Abdullah Ali
Partial differential equations are commonly used in industry and science to model observed phenomena and gain insight regarding phenomena or solve related problems. Recently three dimensional partial differential equations started to become more and more essential and popular. Numerical solution of these problems usually is composed of two steps; discretization with some scheme and solving resulting sparse linear system which is large and usually ill-conditioned. Large size encourages usage of iterative solvers rather that direct solvers due to small memory requirement and short solution times, but iterative solvers mostly fail for ill-conditioned coefficient matrices which is a great discouragement. Preconditioning is a remedy for this problem. Solution of large sparse linear systems take large amounts of time and usually consecutive solution of linear systems is necessary. Parallel computing techniques are used to overcome this problem. Various preconditioning techniques with iterative solutions and their scalability on three different parallel computing platforms are investigated for the solution of two three dimensional partial differential equation related large scale problems which are important for industrial applications and scientific modelling. Results of this investigation are compared against direct solvers and each other.