Abstract. We consider a domain decomposition method for some unsteady heat conduction problem in composite structures. This linear model problem is obtained by homogenization of thin layers of fibres embedded into some standard material. For ease of presentation we consider the case of two space dimensions only. The set of finite element equations obtained by the backward Euler scheme is parallelized in a problem-oriented fashion by some noniterative overlapping domain splitting method, eventually enhanced by inexpensive local iterations to reduce the overlap. We present a detailed convergence analysis of this algorithm which is particularly well appropriate to handle fibre layers of nonlinear material. Special emphasis is to take into account the specific regularity properties of the present mathematical model. Numerical experiments show the reliability of the theoretical predictions.Mathematics Subject Classification. 80A22, 65M.

In solving unsteady problems, domain decomposition methods may be used either for iterative preconditioning each global implicit time-step or directly (noniteratively) within a blockwise implicit time-stepping procedure. In the latter case, the inner boundary values for the subproblems are generated by explicit timeextrapolation. The overlapping variants of this method have been proved to be efficient tools for solving parabolic and first-order hyperbolic problems on modern parallel computers, because they require global communication only once per time-step. The mechanism making this possible is the exponential decay in space of the time-discrete Green's function. We investigate several model problems of convection and convection-diffusion. Favorable optimal and far-reaching estimates of the overlap required have been established in the case of exemplary standard upwind finite-difference schemes. In particular, it has been shown that the overlap for the convection-diffusion problem is the additive function of overlaps for the corresponding convection and diffusion problem to be considered independently. These results have been confirmed with several numerical test examples.

We consider nonstationary convection-dominated flows with stiff source terms. As a unified approach to such problems a combined finite element method in space and time with streamline diffusion is examined numerically. It has good shock-capturing features and is implicit and L-stable. Moreover, being a Galerkin method, it admits residual-based weighted a posteriori error estimates of optimal order. We avoid the use of global stability constants by actually solving the dual problem. This in turn leads to efficient and mathematically rigorous mesh refinement strategies where the streamline diffusion parameter is used for optimizing the resulting adaptive scheme. All theoretical results are substantiated by numerical test examples. For a physical application, we turn to detonation waves at moderate relaxation times in one space dimension. The finite element method well reproduces the Chapman Jouguet speed of detonations and their wave structure. Our error estimator proves accurate and sensitive to parameters, at least on time intervals of moderate length. The adaptive scheme based on this estimator produces fairly accurate results which appear to be the best available with the present strategy. All ingredients of the numerical scheme can be extended in a natural way to higher space dimensions.

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