A painless automatic hp-adaptive strategy for elliptic problems

Abstract

In this work, we introduce a novel hp-adaptive strategy. The main goal is to minimize the complexity and implementational efforts hence increasing the robustness of the algorithm while keeping close to optimal numerical results. We employ a multi-level hierarchical data structure imposing Dirichlet nodes to manage the so-called hanging nodes. The hp-adaptive strategy is based on performing quasi-optimal unrefinements. Taking advantage of the hierarchical structure of the basis functions both in terms of the element size h and the polynomial order of approximation p, we mark those with the lowest contributions to the energy of the solution and remove them. This straightforward unrefinement strategy does not need from a fine grid or complex data structures, making the algorithm flexible to many practical situations and existing implementations. On the other side, we also identify some limitations of the proposed strategy, namely: (a) data structures only support isotropic h-refinements (although p-anisotropic refinements are enabled), (b) we assume certain quasi-orthogonality properties of the basis functions in the energy norm, and (c) in this work, we restrict to symmetric and positive definite problems. We illustrate these and other advantages and limitations of the proposed hp-adaptive strategy with several one-, two-and three-dimensional Poisson examples.

Hierarchical multigrid approaches for the finite cell method on uniform and multi-level hp-refined grids

“…The use of error estimators to guide refinement in the scheme is treated in [54]. A recent publication presents a novel hp-adaptive strategy for multi-level discretiztations [55]. Figure 2 illustrates the manner in which topological components can be activated and deactivated in order to construct the multi-level hp-basis.…”

A Refinement-by-Superposition hp-Method for H(curl)- and H(div)-Conforming Discretizations

“…As such, we opt to extend the refinement-by-superposition (RBS) approach introduced in [15]- [17] for hierarchical basis functions, which demonstrated exponential convergence for scalar problems with C 0 finite elements, to H(curl)and H(div)-conforming finite elements. Additional studies with C 0 finite elements and the RBS hp-method with adaptivity in [18] further motivate extensions of the method to CEM.…”

Efficient multi-level hp-finite elements in arbitrary dimensions

“…For smooth solutions, p-finite element methods converge exponentially with respect to the number of unknowns [1]; for rough solutions, hp-finite elements recover the exponential convergence [2] where element sizes and polynomial degrees gradually decrease towards the solution singularities. The decision of whether to refine h or p can be automated by combining an error estimator and a smoothness indicator (see, e.g., [3,4]). Classical implementations of hp-finite element methods follow a refinement by replacement strategy [5], in which smaller elements substitute elements marked to be refined.…”