24 research outputs found
Variational Multiscale Stabilization and the Exponential Decay of Fine-scale Correctors
This paper addresses the variational multiscale stabilization of standard
finite element methods for linear partial differential equations that exhibit
multiscale features. The stabilization is of Petrov-Galerkin type with a
standard finite element trial space and a problem-dependent test space based on
pre-computed fine-scale correctors. The exponential decay of these correctors
and their localisation to local cell problems is rigorously justified. The
stabilization eliminates scale-dependent pre-asymptotic effects as they appear
for standard finite element discretizations of highly oscillatory problems,
e.g., the poor approximation in homogenization problems or the pollution
effect in high-frequency acoustic scattering
On stability of discretizations of the Helmholtz equation (extended version)
We review the stability properties of several discretizations of the
Helmholtz equation at large wavenumbers. For a model problem in a polygon, a
complete -explicit stability (including -explicit stability of the
continuous problem) and convergence theory for high order finite element
methods is developed. In particular, quasi-optimality is shown for a fixed
number of degrees of freedom per wavelength if the mesh size and the
approximation order are selected such that is sufficiently small and
, and, additionally, appropriate mesh refinement is used near
the vertices. We also review the stability properties of two classes of
numerical schemes that use piecewise solutions of the homogeneous Helmholtz
equation, namely, Least Squares methods and Discontinuous Galerkin (DG)
methods. The latter includes the Ultra Weak Variational Formulation
Static condensation optimal port/interface reduction and error estimation for structural health monitoring
Having the application in structural health monitoring in mind, we propose
reduced port spaces that exhibit an exponential convergence for static
condensation procedures on structures with changing geometries for instance
induced by newly detected defects. Those reduced port spaces generalize the
port spaces introduced in [K. Smetana and A.T. Patera, SIAM J. Sci. Comput.,
2016] to geometry changes and are optimal in the sense that they minimize the
approximation error among all port spaces of the same dimension. Moreover, we
show numerically that we can reuse port spaces that are constructed on a
certain geometry also for the static condensation approximation on a
significantly different geometry, making the optimal port spaces well suited
for use in structural health monitoring
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Plane wave discontinuous Galerkin methods: exponential convergence of the hp-version
We consider the two-dimensional Helmholtz equation with constant coefficients on a domain with piecewise analytic boundary, modelling the scattering of acoustic waves at a sound-soft obstacle. Our discretisation relies on the Trefftz-discontinuous Galerkin approach with plane wave basis functions on meshes with very general element shapes, geometrically graded towards domain corners. We prove exponential convergence of the discrete solution in terms of number of unknowns
Error estimates for Galerkin reduced-order models of the semi-discrete wave equation
Galerkin reduced-order models for the semi-discrete wave equation, that preserve the
second-order structure, are studied. Error bounds for the full state variables are derived
in the continuous setting (when the whole trajectory is known) and in the discrete setting
when the Newmark average-acceleration scheme is used on the second-order semi-discrete
equation. When the approximating subspace is constructed using the proper orthogonal
decomposition, the error estimates are proportional to the sums of the neglected singular
values. Numerical experiments illustrate the theoretical results
Wavenumber explicit convergence analysis for Galerkin discretizations of the Helmholtz equation
We develop a stability and convergence theory for a class of highly indefinite elliptic boundary value problems (bvps) by considering the Helmholtz equation at high wavenumber k as our model problem. The key element in this theory is a novel k-explicit regularity theory for Helmholtz bvps that is based on decomposing the solution into two parts: the first part has the Sobolev regularity properties expected of second order elliptic PDEs but features k-independent regularity constants; the second part is an analytic function for which k-explicit bounds for all derivatives are given. This decomposition is worked out in detail for several types of bvps, namely, the Helmholtz equation in bounded smooth domains or convex polygonal domains with Robin boundary conditions and in exterior domains with Dirichlet boundary conditions. We present an error analysis for the classical hp-version of the finite element method (hp-FEM) where the dependence on the mesh width h, the approximation order p, and the wavenumber k is given explicitly. In particular, under the assumption that the solution operator for Helmholtz problems is polynomially bounded in k, it is shown that quasi optimality is obtained under the conditions that kh/p is sufficiently small and the polynomial degree p is at least O(logk)
The approximate component mode synthesis special finite element method in two dimensions: Parallel implementation and numerical results
A special finite element method based on approximate component mode synthesis (ACMS) was introduced in Hetmaniuk and Lehoucq (2010). ACMS was developed for second order elliptic partial differential equations with rough or highly varying coefficients. Here, a parallel implementation of ACMS is presented and parallel scalability issues are discussed for representative examples. Additionally, a parallel domain decomposition preconditioner (FETI-DP) is applied to solve the ACMS finite element system. Weak parallel scalability results for ACMS are presented for up to 1024 cores. Our numerical results also suggest a quadratic-logarithmic condition number bound for the preconditioned FETI-DP method applied to ACMS discretizations. (C) 2015 Elsevier B.V. All rights reserved