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Mixed \(hp\) finite element methods for problems in elasticity and Stokes flow. (English) Zbl 0855.73075

Summary: We consider the mixed formulation for the elasticity problem and the limiting Stokes problem in \(\mathbb{R}^d\), \(d= 2, 3\). We derive a set of sufficient conditions under which families of mixed finite element spaces are simultaneously stable with respect to the mesh size \(h\) and, subject to a maximum loss of \(O( k^{{d-1} \over 2})\), with respect to the polynomial degree \(k\). We obtain asymptotic rates of convergence that are optimal up to \(O(k^\varepsilon)\) in the displacement/velocity and up to \(O( k^{{{d-1} \over 2}+ \varepsilon})\) in the “pressure”, with \(\varepsilon> 0\) arbitrary (both rates being optimal with respect to \(h\)). Several choices of elements are discussed with reference to properties desirable in the context of the \(hp\)-version.

MSC:

74S05 Finite element methods applied to problems in solid mechanics
76M10 Finite element methods applied to problems in fluid mechanics
76D07 Stokes and related (Oseen, etc.) flows
74B05 Classical linear elasticity
65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
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