Related papers: General framework to construct local-energy soluti…
A method is developed for solving quasilinear convection diffusion problems starting on a coarse mesh where the data and solution-dependent coefficients are unresolved, the problem is unstable and approximation properties do not hold. The…
The energy method can be used to identify well-posed initial boundary value problems for quasi-linear, symmetric hyperbolic partial differential equations with maximally dissipative boundary conditions. A similar analysis of the discrete…
This paper presents a novel multi-scale method for elliptic partial differential equations with arbitrarily rough coefficients. In the spirit of numerical homogenization, the method constructs problem-adapted ansatz spaces with uniform…
This paper concerns a study of the pointwise behaviour of positive solutions to certain quasi-linear elliptic equations with natural growth terms, under minimal regularity assumptions on the underlying coefficients. Our primary results…
We establish sufficient conditions for the local boundedness of weak solutions to a broad class of nonlinear elliptic equations in divergence form, under unbalanced growth conditions on the stress field. Our analysis is carried out in a…
We develop a framework for localized source detection in dynamical systems governed by nonlinear partial differential equations based on first and second-order sensitivity analysis. Building on the standard adjoint formulation, which…
We develop and evaluate a method for learning solution operators to nonlinear problems governed by partial differential equations (PDEs). The approach is based on a finite element discretization and aims at representing the solution…
This work presents the design of nonlinear stabilization techniques for the finite element discretization of Euler equations in both steady and transient form. Implicit time integration is used in the case of the transient form. A…
We consider the initial-boundary value problem for the system of equations describing the flow of compressible isothermal mixture of arbitrary large number of components. The system consists of the compressible Navier-Stokes equations and a…
In the present work, we consider weakly-singular integral equations arising from linear second-order strongly-elliptic PDE systems with constant coefficients, including, e.g., linear elasticity. We introduce a general framework for optimal…
We construct viscosity solutions to the nonlinear evolution equation \eqref{p} below which generalizes the motion of level sets by mean curvature (the latter corresponds to the case $p = 1$) using the regularization scheme as in \cite{ES1}…
This paper deals with the systematic development of structure-preserving approximations for a class of nonlinear partial differential equations on networks. The class includes, for example, gas pipe network systems described by barotropic…
In this work, we study local minimizers of elliptic functionals with strong absorption terms and unbounded, sign-changing sources. These problems naturally interpolate between two classical free boundary problems: Bernoulli-type (cavity)…
We provide a general construction scheme for $\mathcal L^p$-strong Feller processes on locally compact separable metric spaces. Starting from a regular Dirichlet form and specified regularity assumptions, we construct an associated…
This article is a review on basic concepts and tools devoted to a posteriori error estimation for problems solved with the Finite Element Method. For the sake of simplicity and clarity, we mostly focus on linear elliptic diffusion problems,…
This paper develops a new method for analyzing nonlinear diffusion equations of porous--medium type with time-dependent growth rates, based on the \emph{localization of solutions} through an associated Bernoulli-type ordinary differential…
The paper deals with second order parabolic equations on bounded domains with Dirichlet conditions in arbitrary Euclidean spaces. Their interest comes from being models for describing reaction-diffusion processes in several frameworks. A…
We consider the finite element discretization and the iterative solution of singularly perturbed elliptic reaction-diffusion equations in three-dimensional computational domains. These equations arise from the optimality conditions for…
This work addresses the regularity of solutions for a nonlocal diffusion equation over the space of periodic distributions. The spatial operator for the nonlocal diffusion equation is given by a nonlocal Laplace operator with a compactly…
Convergence failure and slow convergence rates are among the biggest challenges with solving the system of non-linear equations numerically. Although mitigated, such issues still linger when using strictly small time steps and…