Novel design and analysis of generalized FE methods based on locally optimal spectral approximations

Abstract

In this paper, the generalized finite element method (GFEM) for solving second order elliptic equations with rough coefficients is studied. New optimal local approximation spaces for GFEMs based on local eigenvalue problems involving a partition of unity are presented. These new spaces have advantages over those proposed in [I. Babuska and R. Lipton, Multiscale Model. Simul., 9 (2011), pp. 373-406]. First, in addition to a nearly exponential decay rate of the local approximation errors with respect to the dimensions of the local spaces, the rate of convergence with respect to the size of the oversampling region is also established. Second, the theoretical results hold for problems with mixed boundary conditions defined on general Lipschitz domains. Finally, an efficient and easy-to-implement technique for generating the discrete A-harmonic spaces is proposed which relies on solving an eigenvalue problem associated with the Dirichlet-to-Neumann operator, leading to a substantial reduction in computational cost. Numerical experiments are presented to support the theoretical analysis and to confirm the effectiveness of the new method.

Wavenumber explicit convergence of a multiscale GFEM for heterogeneous Helmholtz problems

“…Remark 2.4. In contrast to the Dirichlet boundary conditions for local problems in the positive definite case [29], we impose impedance boundary conditions on the artificial interior boundaries for the local Helmholtz problems to guarantee their unique solvability. Such boundary conditions are commonly used as transmission conditions in DDMs for the Helmholtz equation [21].…”

“…Therefore, the exact solution is locally decomposed into two parts, one being the solution of the local Helmholtz problem and another belonging to the generalized harmonic space H B (ω * i ). To approximate the latter part, we follow the lines of [29] to construct a finite-dimensional space that is optimal in an appropriate sense, using the singular vectors of a compact operator involving the partition of unity function. To this end, we first give a novel identity and the resulting Caccioppoli-type inequality for functions in the generalized harmonic space, which plays a crucial role in the analysis of the continuous MS-GFEM.…”

“…Our analysis implies the presence of a resonance effect between the wavenumber and the dimension of the local approximation spaces, which affects the error decay with respect to the oversampling size; see Remark 3.3. In particular, it is shown that the (approximation) error of the method in the high-frequency regime does not always decay with increasing oversampling size, in contrast to the positive definite case [29]. A quasi-optimal global convergence rate for the method is established under the assumption that the size of the subdomains is O(1/k).…”

“…In this paper, we consider the numerical solution of high-frequency heterogeneous Helmholtz problems and deal with the two above focused topics, nonstandard FEM discretizations and efficient solvers, within the unified framework of the Multiscale Spectral Generalized Finite Element Method (MS-GFEM) [1,2,29,28]. Built on the GFEM [30] which constructs the trial space by gluing local approximation spaces together with a partition of unity, the MS-GFEM achieves a high efficiency by building optimal local approximation spaces from selected eigenvectors of carefully designed local eigenvalue problems.…”

“…Furthermore, a proof of the convergence of the discrete eigenvalues as h → 0 to those of the continuous problems is given (h denoting the mesh size), thus also providing guaranteed a posteriori error estimates for the discrete problem for h sufficiently small. As in the case of classical two-level DDMs for the Helmholtz equation, the local approximation spaces and the boundary conditions of the local problems in the method are suitably adapted to the Helmholtz equation, when compared with those for positive definite problems [29,28]; see Remark 2.4. However, although both methods are based on solving some local problems and a global coarse problem, the discrete MS-GFEM can solve the problem in one shot without iteration.…”