Lagrangians for self-adjoint and non-self-adjoint equations

Anti-selfdual Lagrangians: Variational resolutions of non self-adjoint equations and dissipative evolutions

We develop the concept and the calculus of anti-selfdual (ASD) Lagrangians and their “potentials” which seem inherent to many partial differential equations and evolutionary systems. They are natural extensions of gradients of convex functions –hence of self-adjoint positive operators– which usually drive dissipative systems, but also provide representations for the superposition of such gradie...

Self-adjoint differential-algebraic equations

Integrating factors, adjoint equations and Lagrangians

Integrating factors and adjoint equations are determined for linear and nonlinear differential equations of an arbitrary order. The new concept of an adjoint equation is used for construction of a Lagrangian for an arbitrary differential equation and for any system of differential equations where the number of equations is equal to the number of dependent variables. The method is illustrated by...

Non-self-adjoint Differential Operators

We describe methods which have been used to analyze the spectrum of non-self-adjoint differential operators, emphasizing the differences from the self-adjoint theory. We find that even in cases in which the eigenfunctions can be determined explicitly, they often do not form a basis; this is closely related to a high degree of instability of the eigenvalues under small perturbations of the opera...

Non-self-adjoint Linear Systems

We study iterative methods for solving linear systems of the type arising from two-cyclic discretizations of non-self-adjoint two-dimensional elliptic partial differential equations. A prototype is the convection-diffusion equation. The methods consist of applying one step of cyclic reduction, resulting in a "reduced system" of half the order of the original discrete problem, combined with a re...