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We consider additive Schwarz domain decomposition preconditioners for piecewise linear finite element approximations of elliptic PDEs with highly variable coefficients. In contrast to standard analyses, we do not assume that the coefficients can be resolved by a coarse mesh. This situation arises often in practice, for example in the computation of flows in(More)
Coarse spaces are instrumental in obtaining scalability for domain decomposition methods for partial differential equations (PDEs). However, it is known that most popular choices of coarse spaces perform rather weakly in the presence of heterogeneities in the PDE coefficients, especially for systems of PDEs. Here, we introduce in a varia-tional setting a(More)
In this paper we present a new preconditioner suitable for solving linear systems arising from finite element approximations of elliptic PDEs with high-contrast coefficients. The construction of the preconditioner consists of two phases. The first phase is an algebraic one which partitions the degrees of freedom into " high " and " low " permeability(More)
We consider the numerical solution of elliptic partial differential equations with random coefficients. Such problems arise, for example, in uncertainty quantification for groundwater flow. We describe a novel variance reduction technique for the standard Monte Carlo method, called the multilevel Monte Carlo method. The main result is that in certain(More)
We consider a finite element approximation of elliptic partial differential equations with random coefficients. Such equations arise, for example, in uncertainty quantification in subsurface flow modelling. Models for random coefficients frequently used in these applications, such as log-normal random fields with exponential covariance, have only very(More)
Coarse grid correction is a key ingredient in order to have scalable domain decomposition methods. For smooth problems, the theory and practice of such two-level methods is well established, but this is not the case for problems with complicated variation and high contrasts in the coefficients. Stable coarse spaces for high contrast problems are also(More)
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The demand for substantial increases in the spatial resolution of global weather-and climate-prediction models makes it necessary to use numerically efficient and highly scalable algorithms to solve the equations of large scale atmospheric fluid dynamics. For stability and efficiency reasons several of the operational forecasting centres, in particular the(More)
In this paper we consider the problem of computing the spectrum of a Schrödinger operator with discontinuous, periodic potential in two dimensions using Fourier (or planewave expansion) methods. Problems of this kind are currently of great interest in the design of new optical devices to determine band gaps and to compute localised modes in photonic crystal(More)