Abstract
Recently, a multi-level hp-version of the finite element method (FEM) was proposed to ease the difficulties of treating hanging nodes, while providing full hp-approximation capabilities. In the original paper, the refinement procedure made use of a-priori knowledge of the solution. However, adaptive procedures can produce discretizations which are more effective than an intuitive choice of element sizes h and polynomial degree distributions p. This is particularly prominent when a-priori knowledge of the solution is only vague or unavailable. The present contribution demonstrates that multi-level hp-adaptive schemes can be efficiently driven by an explicit a-posteriori error estimator. To this end, we adopt the classical residual-based error estimator. The main insight here is that its extension to multi-level hp-FEM is possible by considering the refined-most overlay elements as integration domains. We demonstrate on several two- and three-dimensional examples that exponential convergence rates can be obtained.
Highlights
The main insight here is that its extension to multi-level hp-finite element method (FEM) is possible by considering the refined-most overlay elements as integration domains
We demonstrate on several two- and three-dimensional examples that exponential convergence rates can be obtained
The finite element method (FEM) has been shown to produce efficient approximations when both refinement in element size h and polynomial degree p are considered
Summary
The finite element method (FEM) has been shown to produce efficient approximations when both refinement in element size h and polynomial degree p are considered (hp-FEM). It is necessary to translate appropriately the multi-level structure to conventional finite elements In this context, we will refer to the set T to be used in (3) as the “partition” of the multi-level mesh Tml. In the standard derivation of the explicit error estimator for the model problem, Green’s theorem is applied to express the energy norm of the approximation error in terms of interior- and boundary-residuals of each element [35,37,40]. The coarsest partition of ensuring C2-continuity of the local shape functions is the set of leaf elements, i.e., the set of elements of any level that are not further refined by a superimposed mesh We present an approach on how to compute the derivative of the numerical solution and the inter-element flux jump in such a recursive setting
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