# Article

**Summary:**

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.