A highly efficient and accurate divergence-free spectral method for curl-curl equation in two and three dimensions

In this paper, we present a fast divergence-free spectral algorithm (FDSA) for the curl-curl problem. Divergence-free bases in two and three dimensions are constructed by using the generalized Jacobi polynomials. An accurate spectral method with exact preservation of the divergence-free constraint point-wisely is then proposed, and its corresponding error estimate is established. We then present a highly efficient solution algorithm based on a combination of matrix-free preconditioned Krylov subspace iterative method and a fully diagonalizable auxiliary problem, which is derived from the spectral discretisations of generalized eigenvalue problems of Laplace and biharmonic operators. We rigorously prove that the dimensions of the invariant subspace of the preconditioned linear system resulting from the divergence-free spectral method with respect to the dominate eigenvalue $1$, are $(N-3)^2$ and $2(N-3)^3$ for two- and three-dimensional problems with $(N-1)^2$ and $2(N-1)^3$ unknowns, respectively. Thus, the proposed method usually takes only several iterations to converge, and astonishingly, as the problem size (polynomial order) increases, the number of iterations will decrease, even for highly indefinite system and oscillatory solutions. As a result, the computational cost of the solution algorithm is only a small multiple of $N^3$ and $N^4$ floating number operations for 2D and 3D problems, respectively. Plenty of numerical examples for solving the curl-curl problem with both constant and variable coefficients in two and three dimensions are presented to demonstrate the accuracy and efficiency of the proposed method.