Scalable tensor-product preconditioners for high-order finite-element methods: Scalar equations

Abstract

We present a tensor-product-based preconditioner for high-order discontinuous-Galerkin (DG) discretizations. The preconditioner is based on approximating the block diagonal of the Jacobian matrix corresponding to element-wise coupling with the sum of tensor products of small one-dimensional matrices. The preconditioner is obtained through an algebraic procedure which minimizes the error between the tensor-product approximation and the exact elemental block Jacobian.Inverting the full elemental block Jacobian requires O(Nd) memory storage and O(N2d) operations, while applying its inverse requires O(Nd) operations per degree of freedom, where N is the solution order and d is the dimension of the problem. Thus, traditional block preconditioners become impractical with increasing dimension, d, and order, N. The cost of forming, storing, or applying the current tensor-product-based preconditioner scales linearly with solution order (O(N)) per degree of freedom for arbitrary number of dimensions. Furthermore, the tensor-product-based preconditioner recovers the exact block-Jacobi preconditioner in the case of constant-coefficient scalar problems on right parallelepiped elements. Numerical results demonstrate the effectiveness of the preconditioner for solving 4D (3D-space+time) DG discretizations of scalar advection-diffusion problems up to 32nd order.