Efficient C1 Bernstein Quasi-Trefftz Discretization for Heterogeneous High-Frequency Helmholtz Problems
Abstract
We develop an efficient C1 Bernstein quasi-Trefftz discretization for heterogeneous high-frequency Helmholtz problems on unstructured triangular meshes. The method first imposes strong C1 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 C1 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 C1 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.
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BibTeX
@misc{kapita2026efficient,
title = {{Efficient $C^1$ Bernstein Quasi-Trefftz Discretization for Heterogeneous High-Frequency Helmholtz Problems}},
author = {Shelvean Kapita},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.05849},
eprint = {2609.05849},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.05849},
}
RIS
TY - UNPB
TI - Efficient C^1 Bernstein Quasi-Trefftz Discretization for Heterogeneous High-Frequency Helmholtz Problems
AU - Shelvean Kapita
PY - 2026
DA - 2026/09/05
PB - arXiv
DO - 10.48550/arxiv.2609.05849
UR - https://arxiv.org/abs/2609.05849
AB - 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.
ER -