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We introduce a framework for the design of finite element methods fortwo-dimensional moving boundary problems with prescribed boundary evolutionthat have arbitrarily high order of accuracy, both in space and in time. At thecore of our approach is the use of a universal mesh: a stationary backgroundmesh containing the domain of interest for all times that adapts to thegeometry of the immersed domain by adjusting a small number of mesh elements inthe neighborhood of the moving boundary. The resulting method maintains anexact representation of the (prescribed) moving boundary at the discrete level,or an approximation of the appropriate order, yet is immune to largedistortions of the mesh under large deformations of the domain. The frameworkis general, making it possible to achieve any desired order of accuracy inspace and time by selecting a preferred and suitable finite-element space onthe universal mesh for the problem at hand, and a preferred and suitable timeintegrator for ordinary differential equations. We illustrate our approach byconstructing a particular class of methods, and apply them to aprescribed-boundary variant of the Stefan problem. We present numericalevidence for the order of accuracy of our schemes in one and two dimensions.

High-Order Finite Element Methods for Moving Boundary Problems with Prescribed Boundary Evolution