Equilibrated flux a posteriori error estimates and adaptivity for nonlinear and doubly degenerate elliptic problems

In this work, we establish a posteriori error analysis for a class of nonlinear and doubly degenerate elliptic equations, including the Stefan problem and both fast and slow diffusion in porous media. Our analysis leverages equilibrated flux reconstructions, yielding guaranteed and fully computable upper bounds on an energy-type error norm with local efficiency. These bounds are fully computable and robust, remaining unaffected by the degree of nonlinearity and degeneracy. The proposed estimators steer an adaptive solver that dynamically switches between nonlinear solvers to optimize iterations. The adaptive algorithm addresses discretization, regularization, and linearization errors. When Newton's method encounters convergence failures, the algorithm adaptively switches to the L-scheme solver. This solver precomputes the stabilization parameter $L>0$ in an offline phase to approximate the Jacobian. The efficiency of the adaptive algorithm is demonstrated through four prototypical examples, showcasing its effective error control and significant computational savings.