Mixed and hybrid finite element methods
Mixed and hybrid finite element methods
On a Data Structure for Adaptive Finite Element Mesh Refinements
ACM Transactions on Mathematical Software (TOMS)
FEMSTER: An object-oriented class library of high-order discrete differential forms
ACM Transactions on Mathematical Software (TOMS)
A compiler for variational forms
ACM Transactions on Mathematical Software (TOMS)
libMesh: a C++ library for parallel adaptive mesh refinement/coarsening simulations
Engineering with Computers
hpGEM---A software framework for discontinuous Galerkin finite element methods
ACM Transactions on Mathematical Software (TOMS)
deal.II—A general-purpose object-oriented finite element library
ACM Transactions on Mathematical Software (TOMS)
Arbitrary-level hanging nodes and automatic adaptivity in the hp-FEM
Mathematics and Computers in Simulation
HiFlow3: a flexible and hardware-aware parallel finite element package
Proceedings of the 9th Workshop on Parallel/High-Performance Object-Oriented Scientific Computing
Algorithms and data structures for massively parallel generic adaptive finite element codes
ACM Transactions on Mathematical Software (TOMS)
SIAM Journal on Scientific Computing
A posteriori error estimates for an optimal control problem of laser surface hardening of steel
Advances in Computational Mathematics
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Finite element methods approximate solutions of partial differential equations by restricting the problem to a finite dimensional function space. In hp adaptive finite element methods, one defines these discrete spaces by choosing different polynomial degrees for the shape functions defined on a locally refined mesh. Although this basic idea is quite simple, its implementation in algorithms and data structures is challenging. It has apparently not been documented in the literature in its most general form. Rather, most existing implementations appear to be for special combinations of finite elements, or for discontinuous Galerkin methods. In this article, we discuss generic data structures and algorithms used in the implementation of hp methods for arbitrary elements, and the complications and pitfalls one encounters. As a consequence, we list the information a description of a finite element has to provide to the generic algorithms for it to be used in an hp context. We support our claim that our reference implementation is efficient using numerical examples in two dimensions and three dimensions, and demonstrate that the hp-specific parts of the program do not dominate the total computing time. This reference implementation is also made available as part of the Open Source deal.II finite element library.