Variational quadratic shape functions for polygons and polyhedra

Solving partial differential equations (PDEs) on geometric domains is an important component of computer graphics, geometry processing, and many other fields. Typically, the given discrete mesh representation should not be altered for simulation purposes. Hence, accurately solving PDEs general meshes a central goal has been considered various operators over last years. While it known that using higher-order basis functions simplicial can substantially improve accuracy convergence, extending these benefits to surface or volume tessellations in efficient fashion remains open problem. Our work proposes variationally optimized piecewise quadratic shape polygons polyhedra, which generalize P 2 elements, exactly reproduce them simplices, inherit their beneficial numerical properties. To mitigate associated cost increased computation time, particularly volumetric meshes, we introduce custom two-level multigrid solver significantly improves computational performance.