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In this paper we study the a priori error estimates for the finite element approximations of parabolic equations with measure data, especially we consider problems with separate measure data in time and space, respectively. The solutions of these kinds of problems exhibit low regularities due to the existence of measure data, this introduces some difficulties in both theoretical and numerical analysis. For both cases we use standard piecewise linear and continuous finite elements for the space discretization and derive the a priori error estimates for the semi-discretization problems, while the backward Euler method is then used for time discretization and a priori error estimates for the fully discrete problems are also derived. Numerical results are provided at the end of the paper to confirm our theoretical finding. © 2012 American Mathematical Society.

Error estimates for finite element approximations of parabolic equations with measure data