  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.
