Time-step heat problem on the mesh: asymptotic behavior and decay rates

Abstract

Abstract In this article, we study the asymptotic behavior and decay of the solution of the fully discrete heat problem. We show basic properties of its solutions, such as the mass conservation principle and their moments, and we compare them to the known ones for the continuous analogue problems. We present the fundamental solution, which is given in terms of spherical harmonics, and we state pointwise and ℓ p {\ell^{p}} estimates for that. Such considerations allow to prove decay and large-time behavior results for the solutions of the fully discrete heat problem, giving the corresponding rates of convergence on ℓ p {\ell^{p}} spaces.