A unified theory for continuous in time evolving finite element space approximations to partial differential equations in evolving domains


We develop a unified theory for continuous in time finite element discretisations of partial differential equations posed in evolving domains including the consideration of equations posed on evolving surfaces and bulk domains as well coupled surface bulk systems. We use an abstract variational setting with time dependent function spaces and abstract time dependent finite element spaces. Optimal a priori bounds are shown under usual assumptions on perturbations of bilinear forms and approximation properties of the abstract finite element spaces. The abstract theory is applied to evolving finite elements in both flat and curved spaces allowing the approximation of parabolic equations in general domains. Numerical experiments are described which confirm the rates of convergence.