MultiShape: a spectral element method, with applications to Dynamic Density Functional Theory and PDE-constrained optimization

Abstract A new numerical framework is developed to solve general nonlinear and nonlocal PDEs on complicated two-dimensional domains. This enables the solution of a wide range of both steady and time-dependent problems on nonstandard geometries, as well as providing the ability to impose nonlinear and nonlocal boundary conditions (typical of those arising in the modelling of physical phenomena) in a flexible and automated way. This spectral element methodology, which we called MultiShape, is compatible with other state-of-the-art numerical methods, such as differential–algebraic equation solvers and optimization algorithms. MultiShape is an open-source Matlab library, in which the numerical implementation is designed to be user-friendly: the problem set-up and computations are done automatically through intuitive operator definitions and notation. Validation tests are presented, before we showcase the power and versatility of MultiShape with three motivating examples in Dynamic Density Functional Theory and PDE-constrained optimization.