Cell-based maximum-entropy approximants

• Smooth, nonnegative maximum-entropy approximants are constructed on unstructured Delaunay meshes.
• Relative entropy measure is minimized, subject to linear reproducing conditions.
• Nodal prior weight functions are formed by taking powers of smooth approximations to the distance function to the boundary of the support.
• Method delivers optimal convergence rates in Sobolev norms for two-dimensional elliptic boundary-value problems.

In this paper, we devise 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 R2R2. 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 are described, and its use to solve two-dimensional elliptic boundary-value problems (Poisson equation and linear elasticity) is demonstrated. The sound accuracy and the optimal rates of convergence of the method in Sobolev norms are established.