Mathematics, Department of

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Error Analysis of the DtN-FEM for the Scattering Problem in Acoustics via Fourier Analysis

Peer reviewed: 
Yes, item is peer reviewed.
Date created: 
2011
Abstract: 

In this paper, we are concerned with the error analysis for the nite elementsolution of the two-dimensional exterior Neumann boundary value problem inacoustics. In particular, we establish an explicit priori error estimates in H1and L2- norms including both the e ect of the truncation of the DtN mappingand that of the numerical discretization. To apply the nite element method(FEM) to the exterior problem, the original boundary value problem is reducedto an equivalent nonlocal boundary value problem via a Dirichlet-to-Neumann(DtN) mapping represented in terms of the Fourier expansion series. We discussessential features of the corresponding variational equation and its modi cationdue to the truncation of the DtN mapping in appropriate function spaces. Nu-merical tests are presented to validate our theoretical results.

Document type: 
Article

Exact Non-Reflecting Boundary Conditions on Perturbed Domains and hp-Finite Elements

Peer reviewed: 
Yes, item is peer reviewed.
Date created: 
2008
Abstract: 

For exterior scattering problems one of the chief difficulties arises from the unboundednature of the problem domain. Inhomogeneous obstacles may require a volumetricdiscretization, such as the Finite Element Method (FEM), and for this approach to be feasiblethe exterior domain must be truncated and an appropriate condition enforced at thefar, artificial, boundary. An exact, non-reflecting boundary condition can be stated usingthe classical DtN-FE method if the Artificial Boundary’s shape is quite specific: circularor elliptical. Recently, this approach has been generalized to permit quite general ArtificialBoundaries which are shaped as perturbations of a circle resulting in the “EnhancedDtN-FE” method. In this paper we extend this method to a two-dimensional FEM featuringhigh-order polynomials in order to realize a high rate of convergence. This is more involvedthan simply specifying high-order test and trial functions as now the scatterer shape andArtificial Boundary must be faithfully represented. This entails boundary elements whichconform (to high order) to the true boundary shapes. As we show, this can be accomplishedand we realize an arbitrary order FEM without spurious reflections.

Document type: 
Article
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High-order Finite Elements on Pyramids: Approximation Spaces, Unisolvency and Exactness

Peer reviewed: 
Yes, item is peer reviewed.
Date created: 
2010-10-21
Abstract: 

We present a family of high-order finite element approximation spaces on a pyramid, and associated unisolvent degrees of freedom. These spaces consist of rational basis functions. We establish conforming, exactness and polynomial approximation properties.

Document type: 
Article
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Convergence Analysis of a Multigrid Algorithm for the Acoustic Single Layer Equation

Peer reviewed: 
Yes, item is peer reviewed.
Date created: 
2012-02-29
Abstract: 

We present and analyze a multigrid algorithm for the acoustic single layer equation in two dimensions. The boundary element formulation of the equation is based on piecewise constant test functions and we make use of a weak inner product in the multigrid scheme as proposed in \cite{BLP94}. A full error analysis of the algorithm is presented. We also conduct a numerical study of the effect of the weak inner product on the oscillatory behavior of the eigenfunctions for the Laplace single layer operator.

Document type: 
Article
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