We study Maxwell's equations in conducting media with perfectly conducting boundary conditions on Lipschitz domains, allowing rough material coefficients and L 2-data. Our first contribution is a direct proof of well-posedness (existence, uniqueness and continuous dependence on data) of the first-order weak formulation. An energy identity is also derived. The argument uses interior-in-time mollification to show uniqueness while avoiding reflection techniques. Existence is via the well-known Galerkin method (cf. Duvaut and Lions [G. Duvaut and J.-L. Lions, Inequalities in Mechanics and Physics, volume 219 of Grundlehren der Mathematischen Wissenschaften, Berlin-New York, Springer-Verlag, 1976. Translated from the French by C. W. John, Eqns. (4.31)-(4.32), p. 346; Thm. 4.1]). For completeness, and to make the paper self-contained, a complete proof has been provided. Our second contribution is a structure-preserving semi-discrete finite element method based on the N & eacute;d & eacute;lec/Raviart-Thomas de Rham complex. The semi-discrete problem is shown to be well-posed. The scheme preserves a discrete Gauss law for all times and satisfies a continuous-in-time energy identity with stability for nonnegative conductivity. With a divergence-free initialization of the magnetic field (via potential reconstruction or constrained L 2 projection), we prove convergence of the semi-discrete solutions to the unique weak solution as the mesh is refined. The analysis mostly relies on projector consistency, weak-* compactness in time-bounded L 2 spaces, and identification of time derivatives in dual spaces. The results at the semi-discrete level are novel.
This paper provides an overview of the new features of the finite element library deal.II, version 9.8, with substantially expanded support for mixed meshes containing tetrahedra, wedges, and pyramids, major improvements to the matrix-free GPU functionality, and many other improvements.
This work investigates a finite element method (FEM) based on the weak imposition of Dirichlet boundary conditions using Nitsche's method for solving singularly perturbed stationary drift-diffusion equations in the quasineutral limit. A priori error analysis is conducted in the underresolved regime, where the mesh is too coarse to effectively resolve boundary layers, showing optimal convergence rates for the proposed method. Numerical experiments additionally demonstrate improved accuracy in terms of global and interior discretization errors as well as contact current evaluation. Further insights into the observed "plateaus" in the convergence plots are obtained by studying a one-dimensional reaction-diffusion problem.
This paper presents a simple and robust method to enforce discrete entropy stability in first-order well-balanced finite volume schemes for systems of balance laws, including non-conservative terms. By leveraging an artificial viscosity technique originally introduced by Tadmor and extended in recent works, the authors propose an entropy-preserving modification applicable to a wide class of three-point schemes. The core contribution lies in the construction of a viscosity coefficient that maintains the well-balanced property while ensuring entropy dissipation at the discrete level. A theoretical framework is developed to guarantee robustness, well-balancing, and entropy stability under a suitable CFL condition. The approach is validated through numerical experiments, including applications to shallow water systems and two-layer flows, demonstrating accuracy and stability.
In this work, we consider a linearized model for fluid flow through a deformable porous elastic solid derived using mixture theory. The model accounts for small but non-negligible solid velocity relative to the fluid, resulting in a coupled system of equations. We focus on the error analysis and implementation of two fully discrete schemes: one using the first-order backward Euler method for time discretization and the other employing the second-order BDF2 scheme. We derive error estimates for both procedures and describe numerical experiments on the corresponding non-dimensional system.
