论文标题
椭圆形PDE系统的自适应BEM，第一部分：虚弱的积分方程的抽象框架
Adaptive BEM for elliptic PDE systems, Part I: Abstract framework for weakly-singular integral equations
论文作者
论文摘要
在目前的工作中，我们考虑了线性二阶强elliptic pde系统带有恒定系数（包括线性弹性）的线性二阶强度PDE系统产生的弱呈弱的积分方程。我们引入了一个通用框架，以最佳的自适应Galerkin BEM收敛。我们确定了基础网格的某些抽象属性，相应的网格再填充策略以及ANSATZ空间，这些空间可以保证以加权误差驱动的自适应算法的最佳代数速率收敛。这些属性得到满足，例如，对于简单网格上的不连续的分段多项式以及某些用于等质分析的ANSATZ空间。技术贡献包括对PDE系统相关的（非本地）边界积分运算符的局部反估计。
In the present work, we consider weakly-singular integral equations arising from linear second-order strongly-elliptic PDE systems with constant coefficients, including, e.g., linear elasticity. We introduce a general framework for optimal convergence of adaptive Galerkin BEM. We identify certain abstract properties for the underlying meshes, the corresponding mesh-refinement strategy, and the ansatz spaces that guarantee convergence at optimal algebraic rate of an adaptive algorithm driven by the weighted-residual error. These properties are satisfied, e.g., for discontinuous piecewise polynomials on simplicial meshes as well as certain ansatz spaces used for isogeometric analysis. Technical contributions include local inverse estimates for the (non-local) boundary integral operators associated to the PDE system.
