A Posteriori Error Estimation for a P1 Finite Element Method Applied to a Biharmonic-Type Problem
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We present a novel a posteriori error estimator for a P1 finite element discretization of a biharmonic-type problem, which consists of finding a function u in H2 such that its Hessian minimizes the L2-norm distance to a given matrix field. This formulation frequently arises in solvers for second-order fully nonlinear equations, such as the Monge-Ampère equation, where the nonlinearity and differential operator are decoupled [1]. To address this fourth-order problem on polygonal domains, we demonstrate its equivalence to a biharmonic problem with the so-called Navier boundary conditions, assuming smooth data. We further reformulate it as a system of coupled second-order equations, which is subsequently solved using mixed finite elements with piecewise linear approximations. We propose a novel a posteriori error estimator that provides a robust framework for adaptive mesh refinement. Numerical experiments demonstrate that i) the L2 error scales quadratically with the mesh size h and the H1 error scales linearly, and, ii) the effectiveness of the refinement strategy. Finally, we showcase the method applying it to the Monge-Ampère equation, and highlighting its broader relevance and computational efficiency.