Jeffrey S. Ovallhttps://works.bepress.com/jeffrey_ovall/Recent works by Jeffrey S. Ovallen-usCopyright (c) 2019 All rights reserved.Fri, 01 Feb 2019 08:00:00 +00003600Analysis of Feast Spectral Approximations Using the DPG Discretizationhttps://works.bepress.com/jeffrey_ovall/14/<p>A filtered subspace iteration for computing a cluster of eigenvalues and its accompanying eigenspace, known as “FEAST”, has gained considerable attention in recent years. This work studies issues that arise when FEAST is applied to compute part of the spectrum of an unbounded partial differential operator. Specifically, when the resolvent of the partial differential operator is approximated by the discontinuous Petrov Galerkin (DPG) method, it is shown that there is no spectral pollution. The theory also provides bounds on the discretization errors in the spectral approximations. Numerical experiments for simple operators illustrate the theory and also indicate the value of the algorithm beyond the confines of the theoretical assumptions. The utility of the algorithm is illustrated by applying it to compute guided transverse core modes of a realistic optical fiber.</p>
Fri, 01 Feb 2019 08:00:00 +0000https://works.bepress.com/jeffrey_ovall/14/ArticlesAn A Posteriori Estimator of Eigenvalue/Eigenvector Error for Penalty-Type Discontinuous Galerkin Methodshttps://works.bepress.com/jeffrey_ovall/13/<p>We provide an abstract framework for analyzing discretization error for eigenvalue problems discretized by discontinuous Galerkin methods such as the local discontinuous Galerkin method and symmetric interior penalty discontinuous Galerkin method. The analysis applies to clusters of eigenvalues that may include degenerate eigenvalues. We use asymptotic perturbation theory for linear operators to analyze the dependence of eigenvalues and eigenspaces on the penalty parameter. We first formulate the DG method in the framework of quadratic forms and construct a companion infinite dimensional eigenvalue problem. With the use of the companion problem, the eigenvalue/vector error is estimated as a sum of two components. The first component can be viewed as a “non-conformity” error that we argue can be neglected in practical estimates by properly choosing the penalty parameter. The second component is estimated a posteriori using auxiliary subspace techniques, and this constitutes the practical estimate.</p>
Thu, 01 Feb 2018 08:00:00 +0000https://works.bepress.com/jeffrey_ovall/13/ArticlesHigh-Order Method for Evaluating Derivatives of Harmonic Functions in Planar Domainshttps://works.bepress.com/jeffrey_ovall/15/<p>We propose a high-order integral equation based method for evaluating interior and boundary derivatives of harmonic functions in planar domains that are specified by their Dirichlet data.</p>
Mon, 01 Jan 2018 08:00:00 +0000https://works.bepress.com/jeffrey_ovall/15/ArticlesFiltered Subspace Iteration for Selfadjoint Operatorshttps://works.bepress.com/jeffrey_ovall/11/<p>We consider the problem of computing a cluster of eigenvalues (and its associated eigenspace) of a (possibly unbounded) selfadjoint operator in a Hilbert space. A rational function of the operator is constructed such that the eigenspace of interest is its dominant eigenspace, and a subspace iteration procedure is used to approximate this eigenspace. The computed space is then used to obtain approximations of the eigenvalues of interest. An eigenvalue and eigenspace convergence analysis that considers both iteration error and dis- cretization error is provided. A realization of the proposed approach for a model second-order elliptic operator is based on a discontinuous Petrov-Galerkin discretization of the resolvent, and a variety of numerical experiments illustrate its performance.</p>
Mon, 18 Sep 2017 07:00:00 +0000https://works.bepress.com/jeffrey_ovall/11/ArticlesSome Remarks on Interpolation and Best Approximationhttps://works.bepress.com/jeffrey_ovall/12/<p>Sufficient conditions are provided for establishing equivalence between best approximation error and projection/interpolation error in finite-dimensional vector spaces for general (semi)norms. The results are applied to several standard finite element spaces, modes of interpolation and (semi)norms, and a numerical study of the dependence on polynomial degree of constants appearing in our estimates is provided.</p>
Wed, 01 Mar 2017 08:00:00 +0000https://works.bepress.com/jeffrey_ovall/12/ArticlesRobust Error Estimates for Approximations of Non-Self-Adjoint Eigenvalue Problemshttps://works.bepress.com/jeffrey_ovall/10/<p>We present new residual estimates based on Kato’s square root theorem for spectral approximations of non-self-adjoint differential operators of convection–diffusion–reaction type. It is not assumed that the eigenvalue/vector approximations are obtained from any particular numerical method, so these estimates may be applied quite broadly. Key eigenvalue and eigenvector error results are illustrated in the context of an hp-adaptive finite element algorithm for spectral computations, where it is shown that the resulting a posteriori error estimates are reliable. The efficiency of these error estimates is also strongly suggested empirically.</p>
Fri, 01 Jul 2016 07:00:00 +0000https://works.bepress.com/jeffrey_ovall/10/ArticlesThe Laplacian and Mean and Extreme Valueshttps://works.bepress.com/jeffrey_ovall/9/<div class="line" id="line-7">The Laplace operator is pervasive in many important mathematical models, and fundamental results such as the mean value theorem for harmonic functions, and the maximum principle for superharmonic functions are well known. Less well known is how the Laplacian and its powers appear naturally in a series expansion of the mean value of a function on a ball or sphere. This result is proven here using Taylor’s theorem and explicit values for integrals of monomials on balls and spheres. This result allows for nonstandard proofs of the mean value theorem and the maximum principle. Connections are also made with the discrete Laplacian arising from finite difference discretization.</div>Jeffrey S. OvallTue, 01 Mar 2016 00:00:00 +0000https://works.bepress.com/jeffrey_ovall/9/ArticlesA Posteriori Eigenvalue Error Estimation for the Schrödinger Operator with the Inverse Square Potentialhttps://works.bepress.com/jeffrey_ovall/3/We develop an a posteriori error estimate of hierarchical type for Dirichlet eigenvalue problems of the form (−∆ + (c/r) 2 )ψ = λψ on bounded domains Ω, where r is the distance to the origin, which is assumed to be in Ω. This error estimate is proven to be asymptotically identical to the eigenvalue approximation error on a family of geometrically-graded meshes. Numerical experiments demonstrate this asymptotic exactness in practice.Wed, 01 Jul 2015 07:00:00 +0000https://works.bepress.com/jeffrey_ovall/3/ArticlesRobust Estimates for hp-Adaptive Approximations of Non-Self-Adjoint Eigenvalue Problemshttps://works.bepress.com/jeffrey_ovall/4/We present new residual estimates based on Kato’s square root theorem for spectral approximations of non-self-adjoint differential operators of convection–diffusion–reaction type. These estimates are incorporated as part of an hp-adaptive finite element algorithm for practical spectral computations, where it is shown that the resulting a posteriori error estimates are reliable. Provided experiments demonstrate the efficiency and reliability of our approach.Stefano Giani et al.Thu, 01 Jan 2015 08:00:00 +0000https://works.bepress.com/jeffrey_ovall/4/ArticlesA Posteriori Estimates Using Auxiliary Subspace Techniqueshttps://works.bepress.com/jeffrey_ovall/2/A posteriori error estimators based on auxiliary subspace techniques for second order elliptic problems in Rd (d ≥ 2) are considered. In this approach, the solution of a global problem is utilized as the error estimator. As the continuity and coercivity of the problem trivially leads to an efficiency bound, the main focus of this paper is to derive an analogous effectivity bound and to determine the computational complexity of the auxiliary approximation problem. With a carefully chosen auxiliary subspace, we prove that the error is bounded above by the error estimate up to oscillation terms. In addition, we show that the stiffness matrix of the auxiliary problem is spectrally equivalent to its diagonal. Several numerical experiments are presented verifying the theoretical results.Wed, 01 Jan 2014 08:00:00 +0000https://works.bepress.com/jeffrey_ovall/2/Articles