Polytopic symmetric interior-penalty approximation of elliptic SPDEs driven by spatial white noise
Abstract
We analyse symmetric interior-penalty discontinuous Galerkin approximations of elliptic stochastic partial differential equations driven by spatial Gaussian white noise on quasi-uniform polytopic meshes. The low regularity of the exact random field precludes the usual SIP consistency argument. We instead restrict the same continuum white-noise functional canonically to the discontinuous polynomial space and separate the error into stochastic forcing-restriction and deterministic discretisation components. For every fixed polynomial degree k≥1, we prove the sharp two-sided estimate |Y−Yh|L2(Ξ;L2(Ω)) ≃ h2−d/2, giving rates h in two dimensions and h1/2 in three dimensions. A finite-dimensional lower bound establishes optimality. The analysis allows polytopic elements with arbitrarily many and arbitrarily small faces, including admissible non-convex elements. In two dimensions we also treat polygons with a single reentrant corner. Although the deterministic approximation exhibits an angle-dependent loss of regularity, the singular resolvent contribution is finite rank, and the sharp first-order stochastic rate is retained. The discontinuous structure yields an exact element-local sampler for the restricted white noise. Numerical experiments on irregular polygonal, non-convex polytopic and reentrant domains support the predicted convergence behaviour.
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BibTeX
@misc{pryer2026polytopic,
title = {{Polytopic symmetric interior-penalty approximation of elliptic SPDEs driven by spatial white noise}},
author = {Tristan Pryer},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.06082},
eprint = {2609.06082},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.06082},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Polytopic symmetric interior-penalty approximation of elliptic SPDEs driven by spatial white noise
AU - Tristan Pryer
PY - 2026
DA - 2026/09/05
PB - arXiv
DO - 10.48550/arxiv.2609.06082
UR - https://arxiv.org/abs/2609.06082
AB - We analyse symmetric interior-penalty discontinuous Galerkin approximations of elliptic stochastic partial differential equations driven by spatial Gaussian white noise on quasi-uniform polytopic meshes. The low regularity of the exact random field precludes the usual SIP consistency argument. We instead restrict the same continuum white-noise functional canonically to the discontinuous polynomial space and separate the error into stochastic forcing-restriction and deterministic discretisation components. For every fixed polynomial degree k≥1, we prove the sharp two-sided estimate |Y−Y_h|_L^2(Ξ;L^2(Ω)) ≃ h^2−d/2, giving rates h in two dimensions and h^1/2 in three dimensions. A finite-dimensional lower bound establishes optimality. The analysis allows polytopic elements with arbitrarily many and arbitrarily small faces, including admissible non-convex elements. In two dimensions we also treat polygons with a single reentrant corner. Although the deterministic approximation exhibits an angle-dependent loss of regularity, the singular resolvent contribution is finite rank, and the sharp first-order stochastic rate is retained. The discontinuous structure yields an exact element-local sampler for the restricted white noise. Numerical experiments on irregular polygonal, non-convex polytopic and reentrant domains support the predicted convergence behaviour.
ER -