发表机构
University of Bath（巴斯大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文针对空间白噪声驱动的椭圆随机偏微分方程，提出并分析了多面体网格上的对称内罚间断伽辽金逼近，证明了最优收敛速率并验证了数值实验。
AI 中文摘要
我们分析了在拟均匀多面体网格上由空间高斯白噪声驱动的椭圆随机偏微分方程的对称内罚间断伽辽金逼近。精确随机场的低正则性排除了通常的SIP一致性论证。我们转而将相同的连续白噪声泛函典范地限制到间断多项式空间，并将误差分离为随机强迫限制和确定性离散化两部分。对于每个固定的多项式次数$k\geq1$，我们证明了尖锐的双侧估计\\[ \\|Y-Y_h\\|_{L^2(\Xi;L^2(\Omega))} \simeq h^{2-d/2}, \\] 在二维中给出速率$h$，在三维中给出速率$h^{1/2}$。有限维下界确立了最优性。该分析允许具有任意多和任意小面的多面体单元，包括允许的非凸单元。在二维中，我们还处理了具有单个凹角的多边形。尽管确定性逼近表现出与角度相关的正则性损失，奇异预解贡献是有限秩的，并且保留了尖锐的一阶随机速率。间断结构为受限白噪声提供了精确的单元局部采样器。在不规则多边形、非凸多面体和凹角域上的数值实验支持了预测的收敛行为。
英文摘要
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 $k\geq1$, we prove the sharp two-sided estimate \[ \|Y-Y_h\|_{L^2(Ξ;L^2(Ω))} \simeq h^{2-d/2}, \] 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.
Comments21 pages, 7 figures
