Stability of sparse space-time finite element discretizations of linear parabolic evolution equations

Abstract

The abstract linear parabolic evolution equation is formulated as a well-posed linear operator equation for which a conforming minimal residual Petrov–Galerkin discretization framework is developed: the approximate solution is defined as the minimizer of a suitable functional residual over the discrete test space, and may be obtained numerically from an equivalent algebraic residual minimization problem. This approximate solution is shown to be well defined and to converge quasi-optimally in the natural norm if the discrete trial and test spaces are stable, i.e., if the discrete inf–sup condition is satisfied with a uniform positive lower bound. For the parabolic operator we devise an abstract criterion for the stability of pairs of space–time trial and test spaces, and construct hierarchic families of trial and test spaces of a sparse space–time tensor-product type that satisfy this criterion. The theory is applied to the concrete example of the diffusion equation and is illustrated numerically.