KAUST DepartmentApplied Mathematics and Computational Science Program
Earth Science and Engineering Program
Numerical Porous Media SRI Center (NumPor)
Physical Sciences and Engineering (PSE) Division
Environmental Science and Engineering Program
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AbstractIn finite element analysis, solving time-dependent partial differential equations with explicit time marching schemes requires repeatedly applying the inverse of the mass matrix. For mass matrices that can be expressed as tensor products of lower dimensional matrices, we present a direct method that has linear computational complexity, i.e., O(N), where N is the total number of degrees of freedom in the system. We refer to these matrices as separable matrices. For non-separable mass matrices, we present a preconditioned conjugate gradient method with carefully designed preconditioners as an alternative. We demonstrate that these preconditioners, which are easy to construct and cheap to apply (O(N)), can deliver significant convergence acceleration. The performances of these preconditioners are independent of the polynomial order (p independence) and mesh resolution (h independence) for maximum continuity B-splines, as verified by various numerical tests. © 2014 Elsevier B.V.
SponsorsThis work was supported in part by the King Abdullah University of Science and Technology (KAUST) Center for Numerical Porous Media and by an Academic Excellence Alliance program award from KAUST's Global Collaborative Research under the title "Seismic wave focusing for subsurface imaging and enhanced oil recovery".