Sparse-grid discretizations of BGG sequences

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.