Knowing thus the Algorithm of this calculus, which I call Differential Calculus, all differential equations can be solved by a common method (Gottfried Wilhelm von Leibniz, 1646–1719).When, several years ago, I saw for the first time an instrument which, when carried, automatically records the number of steps taken by a pedestrian, it occurred to me at once that the entire arithmetic could be subjected to a similar kind of machinery so that not only addition and subtraction, but also multiplication and division, could be accomplished by a suitably arranged machine easily, promptly and with sure results…. For it is unworthy of excellent men to lose hours like slaves in the labour of calculations, which could safely be left to anyone else if the machine was used…. And now that we may give final praise to the machine, we may say that it will be desirable to all who are engaged in computations which, as is well known, are the managers of financial affairs, the administrators of others estates, merchants, surveyors, navigators, astronomers, and those connected with any of the crafts that use mathematics (Leibniz).

This paper is the first part in a series of papers on adaptive finite element methods for parabolic problems. In this paper, an adaptive algorithm is presented and analyzed for choosing the space and time discretization in a finite element method for a linear parabolic problem. The finite element method uses a space discretization with meshsize variable in space and time and a third-order accurate time discretization with timesteps variable in time. The algorithm is proven to be (i) reliable in the sense that the L2-error in space is guaranteed to be below a given tolerance for all timesteps and (ii) efficient in the sense that the approximation error is for most timesteps not essentially below the given tolerance. The adaptive algorithm is based on an a posteriori error estimate which proves (i), and sharp a priori error estimates are used to prove (ii). Analogous results are given for the corresponding stationary (elliptic) problem. In the following papers in this series extensions are made, e.g., to timesteps variable also in space and to nonlinear problems.Key words, adaptive finite element procedures, a priori error estimates, a posteriori error estimates, automatic error control, discontinuous Galerkin method, elliptic problems, parabolic problems

Optimal error estimates are derived for a complete discretization of linear parabolic problems using space-time finite elements. The discretization is done first in time using the discontinuous Galerkin method and then in space using the standard Galerkin method. The underlying partitions in time and space need not be quasi-uniform and the partition in space may be changed from time step to time step. The error bounds show, in particular, that the error may be controlled globally in time on a given tolerance level by controlling the discretization error on each individual time step on the same (given) level, i.e., without error accumulation effects. The derivation of the estimates is based on the orthogonality of the Galerkin procedure and the use of strong stability estimates. The particular and precise form of these error estimates makes it possible to design efficient adaptive methods with reliable automatic error control for parabolic problems in the norms under consideration.Key words, parabolic problem, adaptive finite element method, discontinuous Galerkin method, a priori and a posteriori error estimates GSteborg, 5-412 96 GSteborg, Sweden (Kenneth(C)math.chalmers. 706 Downloaded 11/26/14 to 18.101.24.154. Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php ADAPTIVE FEM FOR PARABOLIC PROBLEMS II 707Galerkin procedures for such problems. We prove this result to be true not only for strictly coercive problems where all eigenvalues of the corresponding linearized elliptic operator are strictly positive but also in cases of slightly negative eigenvalues, which opens possibilities of application, e.g., to reaction-diffusion problems (see [8]). The basic tool to obtain these results is a new stability concept together with the Galerkin orthogonalities built into the finite element method, as discussed further below. See also [10].Our adaptive methods are based on a posteriori error estimates where the error is estimated in terms of a directly computable quantity depending on the local mesh size in time and space, the given data, and the computed solution. The adaptive

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