Development and analysis of an adaptive space–time finite element method for nonlinear parabolic equations of p-Laplace type

We develop and analyze an adaptive spacetime finite element method for nonlinear parabolic equations of p–Laplace type. The model problem is governed by a strongly nonlinear diffusion operator that may be degenerate or singular depending on the exponent p, which typically leads to limited regularity of weak solutions. To address these challenges, we formulate the problem in a unified spacetime variational framework and discretize it using conforming finite element spaces defined on adaptive spacetime meshes. We prove the well-posedness of both the continuous problem and the spacetime discrete formulation, and establish a discrete energy stability estimate that is uniform with respect to the mesh size. Based on residuals in the spacetime domain, we construct a posteriori error estimators and prove their reliability and local efficiency. These results form the foundation for an adaptive spacetime refinement strategy, for which we prove global convergence and quasi-optimal convergence rates without assuming additional regularity of the exact solution. Numerical experiments confirm the theoretical findings and demonstrate that the adaptive spacetime finite element method significantly outperforms uniform refinement and classical time-stepping finite element approaches, particularly for problems exhibiting localized spatial and temporal singularities.