What is known today as the Lee-Friedrichs model is characterized by a self-adjoint operator H on a Hilbert space H, which is the sum of two self-adjoint operators H0 and V , such that H,H0 and V have common domain; H0 has absolutely continuous spectrum (of uniform multiplicity) except for the end-point of the semi-bounded from below spectrum, and one or more eigenvalues which may or may not be ...

By revisiting the basic Godunov approach for linear system of hyperbolic Partial Differential Equations (PDEs) we show that it is hybridizable. As such, it is a natural recipe for us to constructively and systematically establish a unified HDG framework for a large class of PDEs including those of Friedrichs’ type. The unification is fourfold. First, it provides a single constructive procedure ...

In the present work we derive and study a non-linear elliptic PDE coming from the problem of estimation of sound speed inside the Earth. The physical setting of the PDE allows us to pose only a Cauchy problem, and hence is ill-posed. However, we are still able to solve it numerically on a long enough time interval to be of practical use. We used two approaches. The first approach is a finite di...

Developing modern extensions of the Schwarz Alternating Method (SAM) has been a primary focus in the research of domain decomposition during the past ten years. Among the various research eeorts, Generalized Schwarz Alternating Method (GSAM) was an attempt to use the Robin condition !u + (1 ? !) @u @n on those artiicial boundaries to improve the performance of the SAM. Its convergence rate is m...

We consider a simple initial–boundary–value problem for the system of the shallow water equations and its symmetric variant in one space dimension. We discretize the problem in space by the standard Galerkin–finite element method and prove error estimates for the resulting semidiscretizations for quasiuniform and uniform meshes. We also discretize the problem in time using the third–order Shu–O...

Friedrichs’ theory of symmetric positive systems of first-order PDE’s is revisited so as to avoid invoking traces at the boundary. Two intrinsic geometric conditions are introduced to characterize admissible boundary conditions. It is shown that the space in which admissible boundary conditions can be enforced is maximal in a positive cone associated with the differential operator. The equivale...