Space-Time Finite Element Methods for Parabolic Evolution Problems with Non-smooth Solutions
This work addresses the challenge of numerically solving parabolic evolution equations with low-regularity solutions, offering a robust framework for computational scientists and engineers.
The authors develop space-time finite element methods for parabolic problems with non-smooth solutions, providing new a priori estimates and adaptive mesh refinement to maintain convergence rates, and solve the resulting systems with preconditioned GMRES.
We propose consistent locally stabilized, conforming finite element schemes on completely unstructured simplicial space-time meshes for the numerical solution of non-autonomous parabolic evolution problems under the assumption of maximal parabolic regularity. We present new a priori estimates for low-regularity solutions. In order to avoid reduced convergence rates appearing in the case of uniform mesh refinement, we also consider adaptive refinement procedures based on residual a posteriori error indicators. The huge system of space-time finite element equations is then solved by means of GMRES preconditioned by algebraic multigrid.