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"Strouboulis, Theofanis"
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Guaranteed a-posteriori error estimation for finite element solutions of nonstationary heat conduction problems based on their elliptic reconstructions
We deal with the a-posteriori estimation of the error for finite element solutions of nonstationary heat conduction problems with mixed boundary conditions on bounded polygonal domains. The a-posteriori error estimates are constucted by solving stationary “reconstruction” problems, obtained by replacing the time derivative of the exact solution by the time derivative of the finite element solution. The main result is that the reconstructed solutions, or reconstructions, are superconvergent approximations of the exact solution (they are more accurate than the finite element solution) when the error is measured in the gradient or the energy-norm. Because of this, the error in the gradient of the finite element solution can be estimated reliably, by computing its difference from the gradient of its reconstructions. Numerical examples show that “reconstruction estimates” are reliable for the most general classes of solutions which can occur in practical computations.
Journal Article
Partition of unity method for Helmholtz equation: q-convergence for plane-wave and wave-band local bases
In this paper we study the q-version of the Partition of Unity Method for the Helmholtz equation. The method is obtained by employing the standard bilinear finite element basis on a mesh of quadrilaterals discretizing the domain as the Partition of Unity used to paste together local bases of special wave-functions employed at the mesh vertices. The main topic of the paper is the comparison of the performance of the method for two choices of local basis functions, namely a) plane-waves, and b) wave-bands. We establish the q-convergence of the method for the class of analytical solutions, with q denoting the number of plane-waves or wave-bands employed at each vertex, for which we get better than exponential convergence for sufficiently small h, the mesh-size of the employed mesh. We also discuss the a-posteriori estimation for any solution quantity of interest and the problem of quadrature for all integrals employed. The goal of the paper is to stimulate theoretical development which could explain various numerical features. A main open question is the analysis of the pollution and its disappearance as function of h and q.
Journal Article
ADAPTIVE FINITE-ELEMENT METHODS FOR FLOW PROBLEMS IN REGIONS WITH MOVING BOUNDARIES
In this dissertation we study the issues of a-posteriori error estimation and mesh adaptation for a wide class of problems in computational fluid dynamics. The study is divided into two parts: In the first part we provide algorithms for error estimation and mesh enrichment for finite element approximations of viscous incompressible flows, while in the second part we develop techniques for finite element grid optimization for inviscid compressible flow calculations. We first review the basic strategies of a-posteriori error estimation for finite element approximations of elliptic partial differential equations in one and two dimensions. We show that the theory of locally computed a-posteriori error estimates extends naturally to parabolic equations by defining appropriate error estimators computed from the solutions of parabolic local problems. We also develop a theory for a-posteriori residual estimation and a p-method for mesh enrichment in space and time for the Navier-Stokes equations in regions with moving boundaries. We then turn into the study of adaptive finite element methods in compressible gas dynamics, and we develop fast refinement/unrefinement and mesh redistribution methods for the numerical analysis of inviscid compressible flows with particular emphasis to supersonic flows. Finally, we describe a finite element scheme which provides for the computer simulation of the motion of one finite element mesh with respect to another along a smooth interface through which flow occurs. The scheme is designed in order to solve problems of interaction of flows generated by two or more rigid bodies in relative motion in an arbitrary fluid domain. Some applications of this sliding interface/moving grid algorithm to the numerical simulation of supersonic rotor-stator interactions are presented.
Dissertation