Quasi-interpolation for the Helmholtz–Hodge decomposition

Abstract The paper aims at proposing an efficient and stable quasi-interpolation-based method for numerically computing the Helmholtz–Hodge decomposition of a vector field. To this end we first explicitly construct a matrix kernel in a general form from polyharmonic splines such that it includes divergence-free/curl-free/harmonic matrix kernels as special cases. Then, we apply the matrix kernel to vector decomposition via the convolution technique together with the Helmholtz–Hodge decomposition. More precisely, we show that if we convolve a vector field with a scaled divergence-free (curl-free) matrix kernel then the resulting divergence-free (curl-free) convolution sequence converges to the corresponding divergence-free (curl-free) part of the Helmholtz–Hodge decomposition of the field. Finally, by discretizing the convolution sequence via certain quadrature rule, we construct a family of (divergence-free/curl-free) quasi-interpolants for the Helmholtz–Hodge decomposition (defined both in the whole space and over a bounded domain). Corresponding error estimates derived in the paper show that our quasi-interpolation-based method yields convergent approximants to both the vector field and its Helmholtz–Hodge decomposition.