Sharply local pointwise a posteriori error estimates for parabolic problems

We prove pointwise a posteriori error estimates for semi- and fully-discrete finite element methods for approximating the solution $u$ to a parabolic model problem. Our estimates may be used to bound the finite element error $\Vert u-u_h\Vert _{L_\infty(D)}$, where $D$ is an arbitrary subset of the space-time domain of the definition of the given PDE. In contrast to standard global error estimates, these estimators de-emphasize spatial error contributions from space-time regions removed from $D$. Our results are valid on arbitrary shape-regular simplicial meshes which may change in time, and also provide insight into the contribution of mesh change to local errors. When implemented in an adaptive method, these estimates require only enough spatial mesh refinement away from $D$ in order to ensure that local solution quality is not polluted by global effects.