具有诺伊曼边界条件的热方程的自适应时空边界元法
Adaptive space-time BEM for the heat equation with Neumann boundary conditions
AI总结：
研究具有诺伊曼边界条件的热方程的时空BEM，提出加权残差后验误差估计器，用于引导自适应算法，该算法在二维域数值实验中相比均匀细化收敛更快，能在有强奇点时达近最优速率。
AI中文摘要：
我们考虑具有规定初始和诺伊曼数据的热方程的时空边界元法（BEM）。我们提出了一种加权残差后验误差估计器，它是未知BEM误差的上界。假设可能局部细化的网格是抛物缩放棱柱形的，即其元素是时间上的元素\(J\)和空间上的元素\(K\)的张量积\(J\times K\)，且\(|J|\eqsim \text{diam}(K)^2\)。在空间二维域的数值实验中，由导出的估计器引导的自适应算法比均匀细化收敛得快得多，即使在存在强奇点的情况下也能达到近最优速率。
英文摘要：
We consider the space-time boundary element method (BEM) for the heat equation with prescribed initial and Neumann data. We propose a weighted-residual a posteriori error estimator that is an upper bound for the unknown BEM error. The possibly locally refined meshes are assumed to be parabolically scaled prismatic, i.e., their elements are tensor-products $J\times K$ of elements in time $J$ and space $K$ with $|J| \eqsim \text{diam}(K)^2$. In the considered numerical experiments on two-dimensional domains in space, an adaptive algorithm steered by the derived estimator yields significantly faster convergence compared to uniform refinement, achieving near-optimal rates even in the presence of strong singularities.
