We consider the space-time boundary element method (BEM) for the heat equation with prescribed initial and Dirichlet data. We propose a residual-type a posteriori error estimator that is a lower bound and, up to weighted $L_2$-norms of the residual, also an upper bound for the unknown BEM error. The possibly locally refined meshes are assumed to be prismatic, i.e., their elements are tensor-products $J\times K$ of elements in time $J$ and space $K$. While the results do not depend on the local aspect ratio between time and space, assuming the scaling $|J| \eqsim {\rm diam}(K)^2$ for all elements and using Galerkin BEM, the estimator is shown to be efficient and reliable without the additional $L_2$-terms. In the considered numerical experiments on two-dimensional domains in space, the estimator seems to be equivalent to the error, independently of these assumptions. In particular for adaptive anisotropic refinement, both converge with the best possible convergence rate.

翻译：我们认为,热方程式的空时边界要素法(BEM)是带有规定初始和迪里赫莱特数据的热方程式的。 我们提出一种剩余类型的后端误差测算器,该测算器受约束范围较低,最高为2美元,也是未知BEM误差的上限。 可能本地精炼的中间线被假定为具有模糊性, 也就是说, 其元素在时间和空间的元素中是高产品 $J\timets K$, 时间和空间的K$。 虽然结果并不取决于时间和空间之间的局部方位比, 假设是所有元素的缩放 $ ⁇ ⁇ \ eqsim \ rm diam} (K) $2$, 并使用 Galerkin BEM, 该估计器在不增加 $L_ 2 terms 的情况下, 显示是有效和可靠的。 在考虑的两维空间区域的数字实验中, 估计器似乎等同于错误, 与这些假设无关, 特别是适应了对适应异地的精确的精确性调整,, 都与最佳趋同最佳趋近率。