In this paper, we describe a framework to compute expected convergence rates for residuals based on the Calder\'on identities for general second order differential operators for which fundamental solutions are known. The idea is that these rates could be used to validate implementations of boundary integral operators and allow to test operators separately by choosing solutions where parts of the Calder\'on identities vanish. Our estimates rely on simple vector norms, and thus avoid the use of hard-to-compute norms and the residual computation can be easily implemented in existing boundary element codes. We test the proposed Calder\'on residuals as debugging tool by introducing artificial errors into the Galerkin matrices of some of the boundary integral operators for the Laplacian and time-harmonic Maxwell's equations. From this, we learn that our estimates are not sharp enough to always detect errors, but still provide a simple and useful debugging tool in many situations.

翻译：本文提出了一种计算基于卡尔德隆恒等式的残差预期收敛率的框架，该框架适用于已知基本解的一般二阶微分算子。其核心思想在于，这些收敛率可用于验证边界积分算子的实现，并通过选择使卡尔德隆恒等式部分项为零的解来分别测试各算子。我们的估计依赖于简单的向量范数，从而避免了难以计算的范数，且残差计算可轻松集成到现有边界元代码中。我们通过向拉普拉斯算子和时谐麦克斯韦方程组的某些边界积分算子的伽辽金矩阵中引入人为误差，测试了所提出的卡尔德隆残差作为调试工具的有效性。结果表明，虽然我们的估计在检测误差时并非总是足够敏锐，但在多数情况下仍能提供一个简单实用的调试工具。