Mihai Bucataru | Searcharxiv

arXiv2026

Optimal finite element error estimates and Newton convergence for a quasilinear elliptic problem with mixed boundary conditions

The article studies finite element approximations of a quasilinear elliptic heat-conduction problem with inhomogeneous mixed boundary conditions. The conductivity tensor is matrix-valued, anisotropic, possibly nonsymmetric, and dependent on both position and temperature, rendering the problem nonlinear, nonmonotone, and nonpotential. The nonlinear algebraic system arising from the Galerkin discretization is solved using Newton's method, with a posteriori guarantees provided by a computable Newton-Kantorovich criterion and a mesh-dependent stopping rule that ensures that the algebraic error is asymptotically negligible relative to the discretization error. Since the discrete solution need not be unique, we prove that every discrete solution satisfies the optimal convergence rates. The analysis combines mixed-boundary elliptic regularity with an Aubin-Nitsche duality argument adapted to the quasilinear setting. Numerical experiments in two and three dimensions confirm the predicted convergence rates and demonstrate the feasibility of the proposed criterion.

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arXiv2026

Discrete Laplacians on the hyperbolic space -- a comparative study

This paper is concerned with the construction of discrete counterparts of the Laplace-Beltrami operator on Riemannian manifolds that can be effectively used in the numerical solution of partial differential equations. Since existing constructions often lack rigorous convergence guarantees or imply a significant computational effort, we focus on designing operators that are both computationally feasible and supported by convergence results. We consider as a starting point the two-dimensional hyperbolic space $\mathbb{H}^2$, one of the simplest non-Euclidean settings, and develop two variants of discrete finite-difference operator tailored to this constant negatively curved space, both serving as approximations to the (continuous) Laplace-Beltrami operator within the $\mathrm{L}^2$ framework. We prove that the discrete heat equation associated with both operators mentioned above exhibits stability and converges towards the continuous heat-Beltrami Cauchy problem on $\mathbb{H}^2$. Moreover, using techniques inspired from the sharp analysis of discrete functional inequalities, we prove that the solutions of the discrete heat equations corresponding to both variants of discrete Laplacian exhibit an exponential decay asymptotically equal to the one induced by the Poincaré inequality on $\mathbb{H}^2$. Eventually, we illustrate that a discrete Laplacian specifically designed for the geometry of the hyperbolic space yields a more precise approximation and offers advantages from both theoretical and computational perspectives. Furthermore, this discrete operator can be effectively generalized to the three-dimensional hyperbolic space.

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