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We establish the existence of maximal and minimal weak solutions   between ordered pairs of weak sub- and super-solutions for a coupled  system of elliptic equations with quasimonotone nonlinearities on the  boundary. We also formulate a finite difference method to approximate the  solutions and establish the existence of maximal and minimal approximations  between ordered pairs of discrete sub- and super-solutions.   Monotone iterations are formulated for constructing the maximal and minimal  solutions when the nonlinearity is monotone.  Numerical simulations are used to explore existence, nonexistence,  uniqueness and non-uniqueness properties of positive solutions.  When the nonlinearities do not satisfy the monotonicity condition, we prove the existence of weak maximal and minimal solutions using Zorn’s  lemma and a version of Kato’s inequality up to the boundary.  For more information see https://ejde.math.txstate.edu/Volumes/2025/43/abstr.html 

This article develops a unified general framework for designing convergent finite difference and discontinuous Galerkin methods for approximating viscosity and regular solutions of fully nonlinear second order PDEs. Unlike the well-known monotone (finite difference) framework, the proposed new framework allows for the use of narrow stencils and unstructured grids which makes it possible to construct high order methods. The general framework is based on the concepts of consistency and g-monotonicity which are both defined in terms of various numerical derivative operators. Specific methods that satisfy the framework are constructed using numerical moments. Admissibility, stability, and convergence properties are proved,and numerical experiments are provided along with some computer implementation details. For more information see https://ejde.math.txstate.edu/conf-proc/26/f1/abstr.html 

This article analyzes the effect of the penalty parameter used in  symmetric dual-wind discontinuous Galerkin (DWDG) methods for approximating second order elliptic partial differential equations (PDE).  DWDG methods follow from the DG differential calculus framework that defines discrete differential operators used to replace the continuous differential operators when discretizing a PDE. We establish the convergence of the DWDG approximation to a continuous Galerkin approximation as the penalty parameter tends towards infinity. We also test the influence of the regularity of the solution for elliptic second-order PDEs with regards to the relationship between the penalty parameter and the error for the DWDG approximation. Numerical experiments are provided to validate the theoretical results and to investigate the relationship between the penalty parameter and the L^2-error.For more information see https://ejde.math.txstate.edu/conf-proc/26/l1/abstr.html