Finite-Element Preconditioning of G-NI Spectral Methods

Abstract

Several old and new finite-element preconditioners for nodal-based spectral discretizations of −Δu = f in the domain Ω = (−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 P1, Q1, or Q1,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 Nd+1, where N denotes the polynomial degree in each direction

The Interface Control Domain Decomposition Method for Stokes--Darcy Coupling

“…This leads to the so-called Galerkin approach with Numerical Integration (G-NI) [5,4] and to the Spectral Element Method with Numerical Integration (SEM-NI).…”

A collocated C0 finite element method: Reduced quadrature perspective, cost comparison with standard finite elements, and explicit structural dynamics

“…the C 0 -collocation-Galerkin method [50][51][52][53], the differential quadrature method [54,55], the G-NI or SEM-NI methods (Galerkin/spectral element methods with numerical integration) [11,12,56], multidomain spectral or pseudospectral elements [12,31,32,36,57] or hp-FEM with Gauss-Lobatto basis functions [58,59]. Beyond the straightforward implementation of hp-collocation advocated in the present paper, advanced implementation technologies for collocation methods have been developed, which are documented in particular in the spectral element literature [11,12,20,32,48,60,61].…”

Preconditioned Spectral Collocation Method on Curved Element Domains Using the Gordon-Hall Transformation

“…To solve (1) with the spectral collocation method on a curved domain Ω, by the Gordon and Hall transformation [8,9], we transform not only a domain Ω into a square [0, h] × [0, h] but also (1) in Ω into a corresponding secondorder elliptic equation defined in the square domain [0, h] × [0, h] (see section 4). Then, the usual spectral collocation method [2,19] will be employed for the transformed problem, so that the spectral convergence is shown for the transformed problem corresponding to (1).…”

“…Hence the other goal is to precondition such a transformed linear system by a finite element preconditioner [5,16] which is used to get a condition number bounded uniformly. Note that such preconditioners are studied in a square domain in [1], [5] and [16]- [18] for example. Finally, the spectral element collocation method [7] is used on a nonconvex domain for a transformed linear system.…”