Sparse-grid discretizations of BGG sequences

Abstract

We develop the first tensor-product sparse-grid finite element discretizations of Bernstein--Gelfand--Gelfand (BGG) complexes. Alpert wavelets and their first and second primitives yield compatible hierarchies of discontinuous, continuous, and continuously differentiable piecewise polynomials, that serve as one-dimensional ingredients of BGG sparse-grid finite elements. We construct a bounded commuting sparse-grid projection that yields stable discrete potentials for Hessian, elasticity, and div-div complexes. Under mixed Sobolev regularity, we prove sparse-grid approximation estimates with dimension-independent convergence rates in the number of degrees of freedom. Experiments for a three-dimensional incompatibility problem and a two-dimensional biharmonic problem on an annulus illustrate the efficiency of sparse grids relative to full grids.