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A numerical framework is developed to solve various types of PDEs oncomplicated domains, including steady and time-dependent, non-linear andnon-local PDEs, with different boundary conditions that can also includenon-linear and non-local terms. This numerical framework, called MultiShape, isa class in Matlab, and the software is open source. We demonstrate thatMultiShape is compatible with other numerical methods, such asdifferential--algebraic equation solvers and optimization algorithms. Thenumerical implementation is designed to be user-friendly, with most of theset-up and computations done automatically by MultiShape and with intuitiveoperator definition, notation, and user-interface. Validation tests arepresented, before we introduce three examples motivated by applications inDynamic Density Functional Theory and PDE-constrained optimization,illustrating the versatility of the method.

MultiShape: A Spectral Element Method, with Applications to Dynamic Density Functional Theory and PDE-Constrained Optimization