Yves Bourgault

Edward Boey, Yves Bourgault, Thierry Giordano

An element based adaptation method is developed for an anisotropic a posteriori error estimator. The adaptation does not make use of a metric, but instead equidistributes the error over elements using local mesh modifications. Numerical results are reported, comparing with three popular anisotropic adaptation methods currently in use. It was found that the new method gives favourable results for controlling the energy norm of the error in terms of degrees of freedom at the cost of increased CPU usage. Additionally, we considered a new $L^2$ variant of the estimator. The estimator is shown to be conditionally equivalent to the exact $L^2$ error. We provide examples of adapted meshes with the $L^2$ estimator, and show that it gives greater control of the $L^2$ error compared with the original estimator.

Abderrahmane Benfanich, Yves Bourgault, Abdelaziz Beljadid

Standard finite element discretizations of the Richards equation may violate the discrete minimum principle, producing unphysical negative saturations. While existing bound-preserving methods typically rely on computationally expensive fully implicit solvers, we propose a novel semi-implicit finite element framework utilizing a bounded continuous auxiliary variable. Our approach treats the gravity-driven advective term using a linearly implicit technique, which improves the time-step restrictions required by explicit gravity methods near the degenerate limit. We provide rigorous mathematical proofs establishing sufficient geometric and algebraic constraints for discrete positivity and the discrete maximum principle, specifically a local Péclet condition and a discrete row-sum condition. When both conditions are satisfied on weakly acute meshes with mass lumping, our framework ensures that numerical solutions strictly respect physical bounds across highly degenerate conditions and initially dry soil regimes. Comprehensive numerical validation demonstrates the method across multiple flow regimes, including cases where algebraic conditions are satisfied, violated, and recovered through mesh refinement.

Edward Boey, Yves Bourgault, Thierry Giordano

A residual error estimator is proposed for the energy norm of the error for a scalar reaction-diffusion problem and for the monodomain model used in cardiac electrophysiology. The problem is discretized using $P_1$ finite elements in space, and the backward difference formula of second order (BDF2) in time. The estimator for space makes use of anisotropic interpolation estimates, assuming only minimal regularity. Reliability of the estimator is proven under certain mild assumptions on the convergence of the approximate solution. The monodomain model couples a nonlinear parabolic partial differential equation (PDE) with an ordinary differential equation (ODE) and this setting presents challenges theoretically as well as numerically. A space-time adaptation algorithm is proposed to control the global error, using a non-Euclidean metric for mesh adaptation and a simple method to adjust the time step. Numerical examples are used to verify the reliability and efficiency of the estimator, and to test the adaptive algorithm. The potential gains in efficiency of the proposed algorithm compared to methods using uniform meshes is discussed.