We present a new class of self-adaptive higher-order finite element methods ($hp$-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.

EUDML-ID : urn:eudml:doc:271346
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@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/