Efficient $C^1$ Bernstein Quasi-Trefftz Discretization for Heterogeneous High-Frequency Helmholtz Problems

Abstract

We develop an efficient $C^1$ Bernstein quasi-Trefftz discretization for heterogeneous high-frequency Helmholtz problems on unstructured triangular meshes. The method first imposes strong $C^1$ continuity through Bernstein--Bézier smoothness relations on local macro-patches and then reduces the resulting conforming polynomial space by enforcing projected Helmholtz residual constraints. On two-triangle patches, this compresses the local dimension from quadratic growth in the polynomial degree to a trace-sized space of dimension $2p+1$ under the natural rank condition. Variable matrix-valued coefficients are handled through local polynomial projection, allowing coefficient-approximation effects to be separated from discretization error. A graph-residual formulation couples the reduced patch spaces and produces a sparse global system in compressed coordinates. The implementation combines explicit $C^1$ continuation, batched coefficient projection, batched residual construction, and stable local kernel extraction by QR factorization. Numerical experiments on heterogeneous unstructured meshes demonstrate high-order accuracy, roundoff-level $C^1$ conformity, robust high-frequency resolution, and substantial reductions in local and global computational cost. Turning-point and penetrable-scattering examples further illustrate the flexibility of the approach.