Algorithms and Numerical Analysis for Partial Differential Equations

Cornell University, Ithaca NY

Investigators

Abstract

Wahlbin Lars Wahlbin and Alfred Schatz first continue their study of asymptotically exact a posteriori error estimators for the pointwise error in approximating the gradient on each simplex by the finite element method. The problems they consider involve solutions with singularities that pose new difficulties. What makes these problems difficult is that one wants totally local information about errors when, in fact, the solution and its approximation are globally influenced in particular by the singularities. The investigators also look into a priori error estimates for second order elliptic partial differential equations with discontinuous coefficients on nonconvex polygonal domains, allowing meshes that are highly refined. The estimates under consideration justify the use of refined grids in a variety of highly singular problems. Because most refined grids are constructed using self-adaptive codes that may have different methods of choosing a grid, no a priori assumption is made about the distribution of the mesh. The work described above is carried out under the assumption that meshes are shape-regular, but basically nothing else is assumed concerning the meshes. If some further local structure is placed on the mesh (so-called one plus alpha regular meshes), then a limited amount of superconvergence may occur (as shown by Xu and Zhang in a particular situation). The investigators, together with Zhang, apply this idea in greater generality, namely to meshes that are perturbations of those symmetric with respect to a point. The symmetry theory predicts superconvergent points in many practical situations and it is important to determine how this stands up under perturbations. If even more structure is given to the mesh, namely, that it is translation invariant, difference quotients may be used for approximating derivatives. The investigators construct new compact forms of such difference quotients. Finally, the investigators consider finite element methods for visco-elastic problems. The investigators look into basic properties of the finite element approximation method, widely used in engineering and science practice, in industry, in research laboratories as well as in academia. The finite element method builds an approximation to the solution of a differential equation on a region by breaking the region into smaller pieces and approximating the solution on each separate piece, say as a combination of functions that is easy to compute. Pasting together the separate approximations gives a global approximation to the solution. The method has its roots in the airplane industry and started to be used in all branches of science and engineering in tne 1960's. Well over a hundred thousand articles have been written on this method. The investigators' aim in pursuing the fundamental behavior of this method is two-fold: to get a deeper understanding for what the method does, as currently practiced, and to use this understanding to develop new and better methods. A particular aspect that the investigators consider is that of reliably gauging how well the computer method approximates the underlying engineering or scientific model, whose exact solution is of course unknown.