The h-p version of the finite element method with quasiuniform meshes. (English) Zbl 0623.65113

h-p version denotes achieving accuracy by both refining the mesh (decreasing h) and increasing the degree p of the elements. Error estimates of the type \[ \| u-u_{hp}\|_ 1\leq C(h^{\mu - 1}/p^{k-1})\| u\|_ k,\quad \mu =\min (k,p+1) \] are obtained for the model problem \(-\Delta u+u=f\) in \(\Omega\), \(u=g\) on \(\Gamma_ 1\), \(u_ n=b\) on \(\Gamma_ 2\), where \(\Omega\subset {\mathfrak R}^ 2\) is a polygonal domain \((\partial \Omega =\Gamma_ 1\cup \Gamma_ 2)\). Error estimates are also proved when the solution has singularities in the corners of the domain and in the case when g is not in the subspace of finite elements. It is mentioned that all conclusions are valid also for the elasticity problem. A numerical example (plain strain elasticity problem in an L-shaped domain) illustrates the applicability of the developed asymptotic theory.
Reviewer: J.Weisel


65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
74S05 Finite element methods applied to problems in solid mechanics
35J05 Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation
74B05 Classical linear elasticity
65N15 Error bounds for boundary value problems involving PDEs
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