We consider a boundary value problem involving a Riemann-Liouville fractional derivative of order α ∈ (3/2, 2) on the unit interval (0, 1). The standard Galerkin finite element approximation converges slowly due to the presence of singularity term xα−1 in the solution representation. In this work, we develop a simple technique, by transforming it into a second-order two-point boundary value pro...

A uniformly second order method with a local solver based on the piecewise linear discontinuous Galerkin formulation is introduced to solve the eikonal equation with Dirichlet boundary conditions. The method utilizes an interesting phenomenon, referred as the superconvergence phenomenon, that the numerical solution of monotone upwind schemes for the eikonal equation is first order accurate on b...

This paper presents a controller design method for piecewise discrete time linear systems based on a piecewise smooth Lyapunov function. It is shown that the resulting closed loop system is globally stable and the controller can be obtained by solving a set of linear matrix inequalities that is numerically feasible with commercially available software. Copyright © 2002 IFAC

The Cahn-Hilliard equation is discretized by a Galerkin finite element method based on continuous piecewise linear functions in space and discontinuous piecewise constant functions in time. A posteriori error estimates are proved by using the methodology of dual weighted residuals.