Related papers: Discrete comparison principles for quasilinear ell…
Partial differential equations (PDEs) are at the heart of many mathematical and scientific advances. While great progress has been made on the theory of PDEs of standard types during the last eight decades, the analysis of nonlinear PDEs of…
Motivated by many applications in complex domains with boundaries exposed to large topological changes or deformations, fictitious domain methods regard the actual domain of interest as being embedded in a fixed Cartesian background. This…
Nonlinear (systems of) ordinary differential equations (ODEs) are common tools in the analysis of complex one-dimensional dynamic systems. In this paper we propose a smoothing approach regularized by a quasilinearized ODE-based penalty in…
We consider the mathematical analysis and numerical approximation of a system of nonlinear partial differential equations that arises in models that have relevance to steady isochoric flows of colloidal suspensions. The symmetric velocity…
In multi-phase fluid flow, fluid-structure interaction, and other applications, partial differential equations (PDEs) often arise with discontinuous coefficients and singular sources (e.g., Dirac delta functions). These complexities arise…
This paper deals with an existence and uniqueness result of the weak solution for a quasilinear elliptic PDE with nonlinear Robin boundary conditions.This problem is defined on a domain whose boundary is the union of two disjoint…
In this paper we study existence, regularity, and approximation of solution to a fractional semilinear elliptic equation of order $s \in (0,1)$. We identify minimal conditions on the nonlinear term and the source which leads to existence of…
Partial differential equations (PDEs) describing thermodynamically isolated systems typically possess conserved quantities (like mass, momentum, and energy) and dissipated quantities (like entropy). Preserving these conservation and…
We study the continuity of weak solutions for quasilinear elliptic systems with source terms of critical growth arising from a transport-energy structure. The latter occurs frequently in connection with the first balance principles of…
We investigate mathematically a nonlinear approximation type approach recently introduced in [A. Ammar et al., J. Non-Newtonian Fluid Mech., 2006] to solve high dimensional partial differential equations. We show the link between the…
The present paper studies finite element discretizations of second-order elliptic boundary value problems with homogeneous right-hand side and inhomogeneous boundary conditions. We establish discrete spatial decay estimates on element…
The finite element solution of two-dimensional anisotropic diffusion problems is considered. A Delaunay-type mesh condition is developed for linear finite element approximations to satisfy a discrete maximum principle. The condition is…
The validity of the comparison principle in variable coefficient fully nonlinear gradient free potential theory is examined and then used to prove the comparison principle for fully nonlinear partial differential equations which determine a…
We consider semilinear elliptic second-order partial differential inequalities of the form Lu +|u|q-1u < and = Lv +|v|q-1v (*) in the whole space Rn, where n > and = 2, q > 0 and L is a linear elliptic second-order partial differential…
We consider a pointwise tracking optimal control problem for a semilinear elliptic partial differential equation. We derive the existence of optimal solutions and analyze first and, necessary and sufficient, second order optimality…
We provide novel linear combination of unitaries decompositions for a class of discrete elliptic differential operators. Specifically, Poisson problems augmented with periodic, Dirichlet, Neumann, Robin, and mixed boundary conditions are…
The linearizability of differential equations was first considered by Lie for scalar second order semi-linear ordinary differential equations. Since then there has been considerable work done on the algebraic classification of linearizable…
A general sufficient condition for the convergence of subsequences of solutions of non-autonomous, nonlinear difference equations and systems is obtained. For higher order equations the delay sizes and patterns play essential roles in…
We introduce generalised finite difference methods for solving fully nonlinear elliptic partial differential equations. Methods are based on piecewise Cartesian meshes augmented by additional points along the boundary. This allows for…
The linearization of complex ordinary differential equations is studied by extending Lie's criteria for linearizability to complex functions of complex variables. It is shown that the linearization of complex ordinary differential equations…