Higher-Order FEM for a System of Nonlinear Parabolic PDE’s in 2D with A-Posteriori Error Estimates Martin Z´ ıtka 1 , Karel Segeth 2 , and Pavel ˇ Sol´ ın 3 1 Charles University, Prague martan@artax.karlin.mff.cuni.cz 2 Academy of Sciences, Prague segeth@math.cas.cz 3 Rice University, Houston, Texas solin@rice.edu 1 Introduction Initial-boundary value problems for systems of nonlinear parabolic partial differential equations (PDE’s) arise in many important practical applications in electromagnetics, chemistry, modelling of diffusion and heat transfer processes and many other fields 1 . We are concerned with the numerical solution of such problems by the method of lines (MOL) combined with fully automatic hp-adaptive finite element (FE) discretization on unstructured triangular meshes in space. This approach has the potential of reducing the size of discrete problems significantly while preserving the accuracy of results. Until now, automatic hp-adaptivity has been applied almost exclusively to stationary problems (see, e.g., [1, 4, 2] and references therein). It is our aim to extend the promising automatic hp-adaptive strategies for elliptic problems [7, 8] to parabolic equations and in this paper we present two basic steps towards this goal: – Efficient implicit time-adaptive higher-order FE solver PARSYS 2D for systems of non- linear parabolic PDE’s with general boundary conditions for all solution components. – A-posteriori error estimates appropriate for the class of evolutionary problems studied. 2 Definition of the problem Let Ω ⊂ R 2 be a bounded domain with piecewise-polynomial Lipschitz-continuous boundary and J = (0,T ] a finite time interval. We consider a system of N eq nonlin- ear parabolic equations ˙ u(x,t) −∇· (a(u, ∇u, x,t)∇u(x,t)) = f (u, ∇u, x,t) in Ω, t ∈ J, u(x, 0) = v(x) in Ω, u i (x,t)= u D i (x,t) on Γ D i , a i (u, ∇u, x,t)∂u i /∂n = g i (x,t) on Γ N i , (1) (˙ u stands for ∂u/∂t) where u =(u 1 ,...,u Neq ) is the solution, a and f are smooth vector-valued functions and ∂Ω = Γ D i ∪ Γ N i for all i =1,...,N eq . The ∇ (nabla) op- erator is defined as usual, ∇ = ∇ x =(∂/∂x 1 ,∂/∂x 2 ). The vector-valued coefficient a =(a 1 ,...,a Neq ) is bounded, 1 This work was supported by the Grant Agency of the Czech Republic under projects No. GP102/01/D114 and 201/01/1200.