Abstracts for the Conference on Grid Adaptivity in Computational PDEs, Edinburgh, July 96	Next
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Adaptive mesh methods for nonlinear elliptic PDEs with critical Sobolev exponents.

C J Budd

School of Mathematical Sciences,
University of Bath,
Claverton Down,
Bath, BA2 7AY

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Abstract

For certain domains it is known that there exists a critical value such that becomes unbounded as (although remains bounded), and that solutions do not exist if . A question of much interest in analysis is the determination of the value of for a general domain.

If a finite element method using piecewise linear elements on a fixed mesh is used to solve these problems, it can be shown that the resulting solutions converge to when but that they also exist for making a determination of the critical value difficult. We call such latter solutions spurious. At they grow like and appear almost indistinguishable from true solutions. Furthermore, if is greater than, but close to then and as increases this error can be very large.

The accurate resolution of the value of is a challenging problem for an adaptive procedure, which should do two things, firstly it should give an accurate solution when and secondly it should reject the spurious solutions if .

In this talk we will compare the effectiveness of various remeshing techniques for this problem in the special case of a ball where we can exploit symmetry to reduce the complexity of the problem but for which many of the issues of resolving the solution exist. In particular we will look at both static regridding and dynamic regridding methods which vary the mesh as is varied. In both cases the criterion for remeshing depends upon a suitable monitor of the error in the problem.

We show that both methods fail when a monitor is used which is based upon the estimated local truncation error of the solution. In these cases both remeshing methods do not significantly reduce the solution error and also they continue to admit spurious solutions when which appear to be plausible solutions of the problem. The reason for this is that if is the differential operator associated with a linearisation of the differential equation, given by then has eigenvalues which tend to zero as . As a direct consequence the local truncation error does not truly give an indication of the global error in the solution.

By using some formal asymptotic analysis on the underlying problem we can make some estimates of the operator norm of and by using these can show that a reasonable monitor of the error is where is a constant. Using this we can both compute the solution accurately and reject the spurious solutions when static regridding methods (based upon an LUGR adaptive procedure) are used, although we show that moving mesh methods are not effective in this case.

We will also compare the adaptive mesh methods with some extrapolation methods based upon uniform meshes for both spherical and cuboid domains.

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