A numerical investigation on the interplay amongst geometry, meshes, and linear algebra in the finite element solution of elliptic PDEs

作者：Kim Jibum; Sastry Shankar Prasad; Shontz Suzanne M^*

来源：Engineering with Computers, 2012, 28(4): 431-450.

DOI：10.1007/s00366-011-0231-0

In this paper, we study the effect of the choice of mesh quality metric, preconditioner, and sparse linear solver on the numerical solution of elliptic partial differential equations (PDEs). We smooth meshes on several geometric domains using various quality metrics and solve the associated elliptic PDEs using the finite element method. The resulting linear systems are solved using various combinations of preconditioners and sparse linear solvers. We use the inverse mean ratio and radius ratio metrics in addition to conditioning-based scale-invariant and interpolation-based size-and-shape metrics. We employ the Jacobi, SSOR, incomplete LU, and algebraic multigrid preconditioners and the conjugate gradient, minimum residual, generalized minimum residual, and bi-conjugate gradient stabilized solvers. We focus on determining the most efficient quality metric, preconditioner, and linear solver combination for the numerical solution of various elliptic PDEs with isotropic coefficients. We also investigate the effect of vertex perturbation and the effect of increasing the problem size on the number of iterations required to converge and on the solver time. In this paper, we consider Poisson's equation, general second-order elliptic PDEs, and linear elasticity problems.

• 出版日期2012-10