Adaptive hp-FEM with Arbitrary-Level Hanging Nodes for Maxwell’s Equations

Adaptive higher-order finite element methods(hp-FEM)are well known for their potential of exceptionally fast(exponential)convergence.However,most hp-FEM codes remain in an academic setting due to an extreme algorithmic complexity of hp-adaptivity algorithms.This paper aims at simplifying hpadaptivity for H(curl)-conforming approximations by presenting a novel technique of arbitrary-level hanging nodes.The technique is described and it is demonstrated numerically that it makes adaptive hp-FEM more efficient compared to hp-FEM on regular meshes and meshes with one-level hanging nodes.

HANGING NODES IN THE UNIFYING THEORY OF A POSTERIORI FINITE ELEMENT ERROR CONTROL

A unified a posteriori error analysis has been developed in [18, 21-23] to analyze the finite element error a posteriori under a universal roof. This paper contributes to the finite element meshes with hanging nodes which are required for local mesh-refining. The twodimensional 1-irregular triangulations into triangles and parallelograms and their combinations are considered with conforming and nonconforming finite element methods named after or by Courant, Q1, Crouzeix-Raviart, Poisson, Stokes and Navier-Lamé equations Han, Rannacher-Turek, and others for the The paper provides a unified a priori and a posteriori error analysis for triangulations with hanging nodes of degree ≤ 1 which are fundamental for local mesh refinement in self-adaptive finite element discretisations.

ADAPTIVE QUADRILATERAL AND HEXAHEDRAL FINITE ELEMENT METHODS WITH HANGING NODES AND CONVERGENCE ANALYSIS

In this paper we study the convergence of adaptive finite element methods for the gen- eral non-attine equivalent quadrilateral and hexahedral elements on 1-irregular meshes with hanging nodes. Based on several basic ingredients, such as quasi-orthogonality, estimator reduction and D6fler marking strategy, convergence of the adaptive finite element methods for the general second-order elliptic partial equations is proved. Our analysis is effective for all conforming Qm elements which covers both the two- and three-dimensional cases in a unified fashion.

Two-dimensional numerical manifold method with multilayer covers

In order to reach the best numerical properties with the numerical manifold method(NMM),uniform finite element meshes are always favorite while constructing mathematical covers,where all the elements are congruent.In the presence of steep gradients or strong singularities,in principle,the locally-defined special functions can be added into the NMM space by means of the partition of unity,but they are not available to those complex problems with heterogeneity or nonlinearity,necessitating local refinement on uniform meshes.This is believed to be one of the most important open issues in NMM.In this study multilayer covers are proposed to solve this issue.In addition to the first layer cover which is the global cover and covers the whole problem domain,the second and higher layer covers with smaller elements,called local covers,are used to cover those local regions with steep gradients or strong singularities.The global cover and the local covers have their own partition of unity,and they all participate in the approximation to the solution.Being advantageous over the existing procedures,the proposed approach is easy to deal with any arbitrary-layer hanging nodes with no need to construct super-elements with variable number of edge nodes or introduce the Lagrange multipliers to enforce the continuity between small and big elements.With no limitation to cover layers,meanwhile,the creation of an even error distribution over the whole problem domain is significantly facilitated.Some typical examples with steep gradients or strong singularities are analyzed to demonstrate the capacity of the proposed approach.