Quasi-interpolation for the Helmholtz-Hodge decomposition

TL;DR

This paper introduces a stable quasi-interpolation method using matrix kernels derived from polyharmonic splines to efficiently compute the Helmholtz-Hodge decomposition of vector fields with proven convergence and error estimates.

Contribution

It constructs a general matrix kernel framework for vector decomposition and develops a discretized quasi-interpolation method with convergence guarantees.

Findings

Convergent quasi-interpolants for Helmholtz-Hodge decomposition.

Explicit matrix kernel construction from polyharmonic splines.

Error estimates demonstrating the method's accuracy.

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