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501
Domain decomposition for multiscale PDEs
 Numer. Math
"... 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 of ..."
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Cited by 48 (16 self)
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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 heterogeneous porous media, in both the deterministic and (MonteCarlo simulated) stochastic cases. We consider preconditioners which combine local solves on general overlapping subdomains together with a global solve on a general coarse space of functions on a coarse grid. We perform a new analysis of the preconditioned matrix, which shows rather explicitly how its condition number depends on the variable coefficient in the PDE as well as on the coarse mesh and overlap parameters. The classical estimates for this preconditioner with linear coarsening guarantee good conditioning only when the coefficient varies mildly inside the coarse grid elements. By contrast, our new results show that, with a good choice of subdomains and coarse space basis functions, the preconditioner can still be robust even for large coefficient variation inside domains,
Implementation of a boundary element method for high frequency scattering by convex polygons
 ADVANCES IN BOUNDARY INTEGRAL METHODS (PROCEEDINGS OF THE 5TH UK CONFERENCE ON BOUNDARY INTEGRAL METHODS
"... In this paper we consider the problem of timeharmonic acoustic scattering in two dimensions by convex polygons. Standard boundary or finite element methods for acoustic scattering problems have a computational cost that grows at least linearly as a function of the frequency of the incident wave. H ..."
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Cited by 43 (20 self)
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In this paper we consider the problem of timeharmonic acoustic scattering in two dimensions by convex polygons. Standard boundary or finite element methods for acoustic scattering problems have a computational cost that grows at least linearly as a function of the frequency of the incident wave. Here we present a novel Galerkin boundary element method, which uses an approximation space consisting of the products of plane waves with piecewise polynomials supported on a graded mesh, with smaller elements closer to the corners of the polygon. We prove that the best approximation from the approximation space requires a number of degrees of freedom to achieve a prescribed level of accuracy that grows only logarithmically as a function of the frequency. Numerical results demonstrate the same logarithmic dependence on the frequency for the Galerkin method solution. Our boundary element method is a discretisation of a wellknown second kind combinedlayerpotential integral equation. We provide a proof that this equation and its adjoint are wellposed and equivalent to the boundary value problem in a Sobolev space setting for general Lipschitz domains.
Plane wave discontinuous Galerkin methods
, 2007
"... Abstract. We are concerned with a finite element approximation for timeharmonic wave propagation governed by the Helmholtz equation. The usually oscillatory behavior of solutions, along with numerical dispersion, render standard finite element methods grossly inefficient already in mediumfrequency ..."
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Cited by 40 (8 self)
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Abstract. We are concerned with a finite element approximation for timeharmonic wave propagation governed by the Helmholtz equation. The usually oscillatory behavior of solutions, along with numerical dispersion, render standard finite element methods grossly inefficient already in mediumfrequency regimes. As an alternative, methods that incorporate information about the solution in the form of plane waves have been proposed. Among them the ultra weak variational formulation (UWVF) of Cessenat and Despres [O. Cessenat and B. Despres, Application of an ultra weak variational formulation of elliptic PDEs to the twodimensional Helmholtz equation, SIAM J. Numer. Anal., 35 (1998), pp. 255–299.]. We identify the UWVF as representative of a class of Trefftztype discontinuous Galerkin methods that employs trial and test spaces spanned by local plane waves. In this paper we give a priori convergence estimates for the hversion of these plane wave discontinuous Galerkin methods. To that end, we develop new inverse and approximation estimates for plane waves in two dimensions and use these in the context of duality techniques. Asymptotic optimality of the method in a mesh dependent norm can be established. However, the estimates require a minimal resolution of the mesh beyond what it takes to resolve the wavelength. We give numerical evidence that this requirement cannot be dispensed with. It reflects the presence of numerical dispersion. Key words. Wave propagation, finite element methods, discontinuous Galerkin methods, plane waves, ultra weak variational formulation, duality estimates, numerical dispersion AMS subject classifications. 65N15, 65N30, 35J05
Condition Number Estimates for Combined Potential Boundary Integral Operators in Acoustic Scattering
"... We study the classical combined field integral equation formulations for timeharmonic acoustic scattering by a sound soft bounded obstacle, namely the indirect formulation due to BrakhageWerner/Leis/Panič, and the direct formulation associated with the names of Burton and Miller. We obtain lower a ..."
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Cited by 37 (4 self)
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We study the classical combined field integral equation formulations for timeharmonic acoustic scattering by a sound soft bounded obstacle, namely the indirect formulation due to BrakhageWerner/Leis/Panič, and the direct formulation associated with the names of Burton and Miller. We obtain lower and upper bounds on the condition numbers for these formulations, emphasising dependence on the frequency, the geometry of the scatterer, and the coupling parameter. Of independent interest we also obtain upper and lower bounds on the norms of two oscillatory integral operators, namely the classical acoustic single and doublelayer potential operators.
Wavenumberexplicit bounds in timeharmonic scattering
 SIAM J. Math. Anal
"... Abstract. In this paper we consider the problem of scattering of timeharmonic acoustic waves by a bounded sound soft obstacle in two and three dimensions, studying dependence on the wave number in two classical formulations of this problem. The first is the standard variational/weak formulation in ..."
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Cited by 33 (4 self)
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Abstract. In this paper we consider the problem of scattering of timeharmonic acoustic waves by a bounded sound soft obstacle in two and three dimensions, studying dependence on the wave number in two classical formulations of this problem. The first is the standard variational/weak formulation in the part of the exterior domain contained in a large sphere, with an exact DirichlettoNeumann map applied on the boundary. The second formulation is as a second kind boundary integral equation in which the solution is sought as a combined single and doublelayer potential. For the variational formulation we obtain, in the case when the obstacle is starlike, explicit upper and lower bounds which show that the infsup constant decreases like k −1 as the wave number k increases. We also give an example where the obstacle is not starlike and the infsup constant decreases at least as fast as k −2. For the boundary integral equation formulation, if the boundary is also Lipschitz and piecewise smooth, we show that the norm of the inverse boundary integral operator is bounded independently of k if the coupling parameter is chosen correctly. The methods we use also lead to explicit bounds on the solution of the scattering problem in the energy norm when the obstacle is starlike in which the dependence of the norm of the solution on the wave number and on the geometry are made explicit. Key words. Nonsmooth boundary, a priori estimate, infsup constant, Helmholtz equation, oscillatory integral operator AMS subject classifications. 35J05, 35J20, 35J25, 42B10, 78A45 1. Introduction. In
Generalized Robin Boundary Conditions, RobintoDirichlet Maps, and KreinType Resolvent Formulas for Schrödinger Operators on Bounded Lipschitz Domains
 IN PERSPECTIVES IN PARTIAL DIFFERENTIAL EQUATIONS, HARMONIC ANALYSIS AND APPLICATIONS, D. MITREA AND M. MITREA (EDS.), PROCEEDINGS OF SYMPOSIA IN PURE MATHEMATICS, AMERICAN MATHEMATICAL SOCIETY
, 2008
"... We study generalized Robin boundary conditions, RobintoDirichlet maps, and Kreintype resolvent formulas for Schrödinger operators on bounded Lipschitz domains in R n, n � 2. We also discuss the case of bounded C 1,rdomains, (1/2) < r < 1. ..."
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Cited by 31 (11 self)
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We study generalized Robin boundary conditions, RobintoDirichlet maps, and Kreintype resolvent formulas for Schrödinger operators on bounded Lipschitz domains in R n, n � 2. We also discuss the case of bounded C 1,rdomains, (1/2) < r < 1.
RobintoRobin Maps and KreinType Resolvent Formulas for Schrödinger Operators on Bounded Lipschitz Domains
, 2008
"... We study RobintoRobin maps, and Kreintype resolvent formulas for Schrödinger operators on bounded Lipschitz domains in R n, n � 2, with generalized Robin boundary conditions. ..."
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Cited by 28 (10 self)
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We study RobintoRobin maps, and Kreintype resolvent formulas for Schrödinger operators on bounded Lipschitz domains in R n, n � 2, with generalized Robin boundary conditions.
An adaptive perfectly matched layer technique for timeharmonic scattering problems
 SIAM J. Numer. Anal
"... Abstract. An adaptive perfectly matched layer (PML) technique for solving the time harmonic electromagnetic scattering problems is developed. The PML parameters such as the thickness of the layer and the fictitious medium property are determined through sharp a posteriori error estimates. Combined w ..."
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Cited by 27 (9 self)
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Abstract. An adaptive perfectly matched layer (PML) technique for solving the time harmonic electromagnetic scattering problems is developed. The PML parameters such as the thickness of the layer and the fictitious medium property are determined through sharp a posteriori error estimates. Combined with the adaptive finite element method, the adaptive PML technique provides a complete numerical strategy to solve the scattering problem in the framework of FEM which produces automatically a coarse mesh size away from the fixed domain and thus makes the total computational costs insensitive to the thickness of the PML absorbing layer. Numerical experiments are included to illustrate the competitive behavior of the proposed adaptive method. 1.