Jeffrey S. Ovall

On estimators for eigenvalue/eigenvector approximations

We consider a large class of residuum based a posteriori eigen-value/eigenvector estimates and present an abstract framework for proving their asymptotic exactness. Equivalence of the estimator and the error is also established. To demonstrate the strength of our abstract approach we present a detailed study of hierarchical error estimators for Laplace eigenvalue problems in planar polygonal regions. To this end we develop new error analysis for the Galerkin approximation which avoids the use of the strengthened Cauchy-Schwarz inequality and the saturation assumption, and gives reasonable and explicitly computable upper bounds on the discretization error. A brief discussion is also given concerning the design of estimators which are in the same spirit, but are based on different a posteriori techniques-notably, those of gradient recovery type.

Asymptotically exact functional error estimators based on superconvergent gradient recovery

The use of dual/adjoint problems for approximating functionals of solutions of PDEs with great accuracy or to merely drive a goal-oriented adaptive refinement scheme has become well-accepted, and it continues to be an active area of research. The traditional approach involves dual residual weighting (DRW). In this work we present two new functional error estimators and give conditions under which we can expect them to be asymptotically exact. The first is of DRW type and is derived for meshes in which most triangles satisfy an -approximate parallelogram property. The second functional estimator involves dual error estimate weighting (DEW) using any superconvergent gradient recovery technique for the primal and dual solutions. Several experiments are done which demonstrate the asymptotic exactness of a DEW estimator which uses a gradient recovery scheme proposed by Bank and Xu, and the effectiveness of refinement done with respect to the corresponding local error indicators.

Function, Gradient, and Hessian Recovery Using Quadratic Edge-Bump Functions

An approximate error function for the discretization error on a given mesh is obtained by projecting (via the energy inner product) the functional residual onto the space of continuous, piecewise quadratic functions which vanish on the vertices of the mesh. Conditions are given under which one can expect this hierarchical basis error estimator to give efficient and reliable function recovery, asymptotically exact gradient recovery, and convergent Hessian recovery in the square norms. One does not find similar function recovery results in the literature. The analysis given here is based on a certain superconvergence result which has been used elsewhere in the analysis of gradient recovery methods. Numerical experiments are provided which demonstrate the effectivity of the approximate error function in practice.

A high-order integral algorithm for highly singular PDE solutions in Lipschitz domains

We present a new algorithm, based on integral equation formulations, for the solution of constant-coefficient elliptic partial differential equations (PDE) in closed two-dimensional domains with non-smooth boundaries; we focus on cases in which the integral-equation solutions as well as physically meaningful quantities (such as, stresses, electric/magnetic fields, etc.) tend to infinity at singular boundary points (corners). While, for simplicity, we restrict our discussion to integral equations associated with the Neumann problem for the Laplace equation, the proposed methodology applies to integral equations arising from other types of PDEs, including the Helmholtz, Maxwell, and linear elasticity equations. Our numerical results demonstrate excellent convergence as discretizations are refined, even around singular points at which solutions tend to infinity. We demonstrate the efficacy of this algorithm through applications to solution of Neumann problems for the Laplace operator over a variety of domains—including domains containing extremely sharp concave and convex corners, with angles as small as π/100 and as large as 199π/100.

Well conditioned boundary integral equations for two-dimensional sound-hard scattering problems in domains with corners

We present several well-posed, well-conditioned direct and indirect integral equation formulations for the solution of two-dimensional acoustic scattering problems with Neumann boundary conditions in domains with corners. We focus mainly on Direct Regularized Combined Field Integral Equations (DCFIE-R) formulations whose name reflects that (1) they consist of combinations of direct boundary integral equations of the second-kind and first-kind integral equations which are preconditioned on the left by coercive boundary single-layer operators, and (2) their unknowns are physical quantities, i.e the total field on the boundary of the scatterer. The DCFIE-R equations are shown to be uniquely solvable in appropriate function spaces under certain assumptions on the coupling parameter. Using Calderon’s identities and the fact that the unknowns are bounded in the neighborhood of the corners, the integral operators that enter the DCFIE-R formulations are recast in a form that involves integral operators that are expressed by convergent integrals only. The polynomially-graded mesh quadrature introduced by Kress [30] enables the high-order resolution of the weak singularities of the kernels of the integral operators and the singularities in the derivatives of the unknowns in the vicinity of the corners. This approach is shown to lead to an efficient, high-order Nystrom method capable of producing solutions of sound-hard scattering problems in domains with corners which require small numbers of Krylov subspace iterations throughout the frequency spectrum. We present a variety of numerical results that support our claims.

A posteriori error estimation of hierarchical type for the Schrödinger operator with inverse square potential

We develop an a posteriori error estimate for mixed boundary value problems of the form

A Posteriori Estimates Using Auxiliary Subspace Techniques

(-\Delta +\fancyscript{V})u=f

Error control for hp-adaptive approximations of semi-definite eigenvalue problems

(-Δ+V)u=f, where the potential