Cell-based maximum-entropy approximants

DANIEL MILLÁN; MARINO ARROYO; N. SUKUMAR

Tampa, Florida

Workshop; USACM Workshop on Meshfree Methods for Large-Scale Computational Science and Engineering; 2014

United States Association for Computational Mechanics

In this talk, we present cell-based maximum-entropy (max-ent) basis functions that are used in a Galerkin method for the solution of partial differential equations. The motivation behind this work is the construction of smooth approximants with controllable support on unstructured meshes. In the variational scheme to obtain max-ent basis functions, the nodal prior weight function is constructed from an approximate distance function to a polygonal curve in R2. More precisely, we take powers of the composition of R-functions via Boolean operations. The basis functions so constructed are nonnegative, smooth, linearly complete, and compactly-supported in a neighbor-ring of segments that enclose each node. The smoothness is controlled by two positive integer parameters: the normalization order of the approximation of the distance function and the power to which it is raised. The properties and mathematical foundations of the new compactly-supported approximants will be described, and its use to solve elliptic two-dimensional boundary-value problems (Poisson equation and linear elasticity) will be demonstrated. The sound accuracy and the optimal rates of convergence of the method in Sobolev norms will be established.