Mollified Approximants for Finite Elements and Collocation over Polytopic Discretisation
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We introduce the smooth high-order mollified basis functions for solving boundary value problems over partitions consisting of convex polytopes. The proposed construction begins by assuming an independent local polynomial approximant of arbitrary order over each polytope. Basis functions are then obtained by convolving these local approximants with a mollifier that is smooth, compactly supported and has unit volume. The approximation properties of the so-obtained basis functions depend on the order of the local approximants and the smoothness of the mollifier. Convolution integrals are evaluated first by computing the Boolean intersection between the mollifier support and the polytopes and then repeatedly applying the divergence theorem to simplify the integrals. The support of a basis function is the Minkowski sum of the respective polytope and the mollifier. We employ the mollified basis functions to discretise the physical governing equations using the collocation method [1] or their weak form using the finite element method [2]. To accommodate simulations involving complex geometries, we use an implicit description of the boundary, i.e., a level set immersed in a non-boundary fitting polytopic background mesh. The studied numerical examples confirm the optimal convergence of the developed approximation scheme for Poisson, elasticity, and high-order PDEs such as biharmonic equation and plate bending.