The Approximation Theory for the P-Version of the Finite Element Method. I.
MARYLAND UNIV COLLEGE PARK LAB FOR NUMERICAL ANALYSIS
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In its standard mathematical formulation, the finite element method is a particular kind of Ritz-Galerkin procedure in which the approximating finite-dimensional subspaces are composed of piecewise polynomials defined on a partition of the given domain into convex subdomains. Since the convergence of such methods is obtained by increasing the dimension of these subspaces in some manner, one observes that there are basically two ways this can be done. The first way is the traditional approach obtained by fixing the degree p of the piecewise polynomials at some value p 1,2,3 and decreasing the mesh size h in order to achieve convergence this is known as the h-version of the finite element method. The second way, referred to as the p-version of the finite element method, is to fix the mesh and increase the degree p in order to reduce the approximation error. Clearly, a combination of the two is also possible. While the h-version has been extensively investigated in the mathematical literature and hav been widely used in engineering applications for many years, the development of the p-version has taken place only recently. It is now recognized that for many problems of engineering and scientific interest, the p-version offers a number of advantages over the h-version both in the quality of approximation and in the cost of computation.
- Numerical Mathematics