Finite-Element Preconditioning of G-NI Spectral Methods

Several old and new finite-element preconditioners for nodal-based spectral discretizations of $-\Delta u=f$ in the domain $\Omega=(-1,1)^d$ ($d=2$ or 3), with Dirichlet or Neumann boundary conditions, are considered and compared in terms of both condition number and computational efficiency. The computational domain covers the case of classical single-domain spectral approximations (see [C. Canuto et al., Spectral Methods. Fundamentals in Single Domains, Springer, Heidelberg, 2006]), as well as that of more general spectral-element methods in which the preconditioners are expressed in terms of local (upon every element) algebraic solvers. The primal spectral approximation is based on the Galerkin approach with numerical integration (G-NI) at the Legendre–Gauss–Lobatto (LGL) nodes in the domain. The preconditioning matrices rely on either $\mathbb{P}_1$, $\mathbb{Q}_1$, or $\mathbb{Q}_{1,NI}$ (i.e., with numerical integration) finite elements on meshes whose vertices coincide with the LGL nodes used for the spectral approximation. The analysis highlights certain preconditioners, which yield the solution at an overall cost proportional to $N^{d+1}$, where N denotes the polynomial degree in each direction.