We present a new class of self-adaptive higher-order finite element methods (-FEM) which are free of analytical error estimates and thus work equally well for virtually all PDE problems ranging from simple linear elliptic equations to complex time-dependent nonlinear multiphysics coupled problems. The methods do not contain any tuning parameters and work reliably with both low- and high-order finite elements. The methodology was used to solve various types of problems including thermoelasticity, microwave heating, flow of thermally conductive liquids etc. In this paper we use a combustion problem described by a system of two coupled nonlinear parabolic equations for illustration. The algorithms presented in this paper are available under the GPL license in the form of a modular C++ library HERMES.
@article{702873,
title = {Space-time adaptive $hp$-FEM: Methodology overview},
booktitle = {Programs and Algorithms of Numerical Mathematics},
series = {GDML\_Books},
publisher = {Institute of Mathematics AS CR},
address = {Prague},
year = {2008},
pages = {185-200},
zbl = {05802259},
url = {http://dml.mathdoc.fr/item/702873}
}
Šolín, Pavel; Segeth, Karel; Doležel, Ivo. Space-time adaptive $hp$-FEM: Methodology overview, dans Programs and Algorithms of Numerical Mathematics, GDML_Books, (2008), pp. 185-200. http://gdmltest.u-ga.fr/item/702873/