On the Stabilizer‐Free Immersed Weak Galerkin FEM: A Posteriori Error Estimation in a Non‐Conforming Framework

ABSTRACT This work focuses on a posteriori error estimation within a non‐conforming framework for the stabilizer‐free immersed weak Galerkin finite element method (IWG‐FEM) applied to elliptic and parabolic problems with non‐smooth coefficients. With an appropriate adaptation of the Helmholtz decomposition of the error, an optimal a posteriori error bound is derived in the weighted ‐seminorm to address elliptic problems with non‐smooth coefficients. For parabolic problems, we analyze a fully discrete scheme that combines the implicit backward Euler time‐stepping method with the non‐conforming IWG‐FEM. A reliable a posteriori error indicator is derived using the time‐dependent Helmholtz decomposition in the ‐norm in time and the weighted ‐seminorm in space without directly relying on the energy argument. Finally, numerical experiments on various test problems confirm the convergence behavior of the proposed error indicators.