Abstract
SummaryIn this paper, a high‐order finite element method for partial differential equations on smooth surfaces is proposed. The surface is defined as the intersection of a rectangular cuboid and an implicitly defined surface. Therefore, the surface of interest may not be closed. The main novel contribution in this work is the incorporation of an exact geometry description of surfaces with boundary into the finite element method. To this end, a piecewise planar triangulation is mapped onto the surface of interest by making use of the implicit surface definition. The mapping uses predefined search directions and can, therefore, be tailored to consider boundaries. High‐order hierarchical shape functions are utilized for the field approximation. They are defined on a reference triangle in the usual way. The proposed method is easy to implement and bypasses the need for a high‐order geometry description. Furthermore, due to the exact geometry, the imposition of Dirichlet boundary conditions, source terms, and mesh refinement are easy to carry out.
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