A posteriori error estimator is derived for an elliptic interface problem in the fictitious domain formulation with distributed Lagrange multiplier considering a discontinuous Lagrange multiplier finite element space. A posteriori error estimation plays a pivotal role in assessing the accuracy and reliability of computational solutions across various domains of science and engineering. This study delves into the theoretical underpinnings and computational considerations of a residual-based estimator. Theoretically, the estimator is studied for cases with constant coefficients which jump across an interface as well as generalized scenarios with smooth coefficients that jump across an interface. Theoretical findings demonstrate the reliability and efficiency of the proposed estimators under all considered cases. Numerical experiments are conducted to validate the theoretical results, incorporating various immersed geometries and instances of high coefficients jumps at the interface. Leveraging an adaptive algorithm, the estimator identifies regions with singularities and applies refinement accordingly. Results substantiate the theoretical findings, highlighting the reliability and efficiency of the estimators. Furthermore, numerical solutions exhibit optimal convergence properties, demonstrating resilience against geometric singularities or coefficients jumps.

翻译：针对采用分布拉格朗日乘子并考虑间断拉格朗日乘子有限元空间的虚拟域公式椭圆界面问题，推导了后验误差估计器。后验误差估计在评估科学与工程各领域计算解的精度与可靠性方面具有关键作用。本研究深入探讨了基于残差的估计器的理论基础与计算考量。理论上，该估计器针对系数在界面处跳跃的常系数情形及系数在界面处跳跃的光滑系数广义场景进行了研究。理论结果表明，在所有考虑情形下所提估计器均具有可靠性与高效性。通过数值实验验证了理论结果，实验涵盖了多种浸入几何构型及界面处高系数跳跃的实例。借助自适应算法，估计器能识别具有奇异性的区域并相应实施网格细化。结果证实了理论发现，凸显了估计器的可靠性与高效性。此外，数值解展现出最优收敛特性，表现出对几何奇异性或系数跳跃的强健适应性。