Error estimates for the combinedh andp versions of the finite element method

Abstract

Summary.In the h-version of the finite element method, convergence is achieved by refining the mesh while keeping the degree of the elements fixed. On the other hand, the p-version keeps the mesh fixed and increases the degree of the elements. In this paper, we prove estimates showing the simultaneous dependence of the order of approximation on both the element degrees and the mesh. In addition, it is shown that a proper design of the mesh and distribution of element degrees lead to a better than polynomial rate of convergence with respect to the number of degrees of freedom, even in the presence of corner singularities. Numerical results comparing the h-version, p-version, and combined h-p-version for a one dimensional problem are presented.

The post‐processing approach in the finite element method—part 1: Calculation of displacements, stresses and other higher derivatives of the displacements

“…possible by suitable refinement about the corner points to obtain arbitrarily large orders of convergence with respect to the number of degrees-of-freedom (see (3] integration by parts gives…”

On the rates of convergence of the finite element method

“…The proof of this theorem is given in [16]. The significance of this theorem is that it joins theorems 3 and 4, making the constant C independent of both p and T. An inverse theorem exists but is not known as yet.…”

The $h-p$ version of the finite element method with quasiuniform meshes

“…We now define (7.19) *a(*) = *i(x) + * 2 (r), * 2 (*) = *i(*)-tf 2 (x) so that (7.18) yields f = h x {x) (7.206) Here ^(JC) is unique, *|ii(0) = 0, while i|f 2 (jc) is unique up to the constant K such that *|i 2 (0) = 2ÜC. We first analyze (7.20a Then it is easy to see that FÊ^(T), F = ƒ x on 7 l3 F = 0 on 7 2 , 73 and because of (7.28), (7.31) and ( The mapping The theorem is therefore proven in the case that ƒ = 0 on three sides of S and hence it holds also if ƒ is gênerai but ƒ = 0 at the vertices ABCD. It remams to prove that in the gênerai case there exist # € £Pp(S) such that <ï > has the same traces at ABCD as ƒ and ll*lli,5« c |io||/i|| ia J-…”

“…The first theoretical paper appeared in 1981 (see [6]). See also [2,5,7,10,11,14] for most recent results. For the numerical, computational, implementational and engineering aspects of the h-p version we refer to [3,[21][22][23][24],…”