Structure-preserving quasi-interpolation for vector-valued function with multiple physical constraints

Abstract

We develop a unified theory of vectorial quasi-interpolation that exactly preserves intrinsic physical structures characterized by systems of linear constant-coefficient differential operators, including divergence-free and curl-free constraints as special cases. The key ingredient is a family of matrix-valued kernels constructed from an orthogonal projector in the frequency domain, whose columns analytically satisfy the prescribed constraints. The resulting quasi-interpolant is a weighted combination of kernel translates with sampled function values as coefficients, and therefore requires no constrained optimization. Simultaneous error estimates for the approximand and its derivatives are established within a bias-variance framework. For numerical vector-field decomposition, we introduce a generalized Helmholtz--Hodge decomposition that splits a smooth vector field into two components with distinct physical structures, and develop corresponding structure-preserving quasi-interpolation schemes and error estimates. Explicit kernels are derived for several representative constraints, including steady-state acoustic constraints, combined divergence-curl constraints, coupled divergence-free constraints, and second-order Saint-Venant compatibility conditions. Numerical experiments confirm the theoretical error estimates and exact structure preservation, demonstrate accurate identification of sources and sinks, and show three-dimensional reconstruction of generalized Helmholtz--Hodge components for general smooth vector fields without prior constraint assumptions.