Abstract: Space and time discretisations of parabolic differential equations with dynamic boundary conditions are studied in a weak formulation that fits into the standard abstract formulation of parabolic problems, just that the usual inner product is replaced by an suitable inner product on a problem specific Hilbert space. The class of parabolic equations considered includes linear problems with time- and space-dependent coefficients and semi-linear problems such as reaction-diffusion on a surface coupled to diffusion in the bulk. The spatial discretisation by finite elements is studied in the proposed framework, with particular attention to the error analysis of the Ritz map for the elliptic bilinear form in relation to the inner product, both of which contain boundary integrals. We further consider mass lumping, which enables us to use exponential integrators and bulk-surface splitting for time integration, which yields surprising numerical results.
Abstract: Space-fractional diffusion processes were observed in the last decades in many real-life situations. To simulate such phenomena, we first need an appropriate model. It turned out that a classical topic, the so-called fractional-order calculus can be applied here. We also discuss some approaches for the numerical solution of the corresponding problems, both finite difference and finite element methods. The matrix transformation method is investigated in details. We also mention some open questions of the topic.