Efficient Parallel Preconditioners for High-Order Finite Element Discretizations of H(grad) and H(curl) Problems

In this paper, we study preconditioning techniques for both H(grad) and H(curl) problems. For an H(grad) elliptic problem discretized by high-order finite elements with a hierachical basis of k-th order, we design and analyse an parallel AMG preconditioner based on a two-level method and a block Gauss–Seidel smoothing technique. For an H(curl) elliptic problem discretized by high-order edge finite elements, we design a parallel solver based on an auxiliary space preconditioner. Numerical experiments show that the number of iteration for the corresponding PCG methods does not depend on the mesh size, and depend weakly on the order of the finite element space and jumps of the coefficients.