In this paper, we develop the locally divergence-free discontinuous Galerkin method for numerically solving the Maxwell equations. The distinctive feature of the method is the use of approximate solutions that are exactly divergence-free inside each element. As a consequence, this method has a smaller computational cost than that of the discontinuous Galerkin method with standard piecewise poly...

A simple method for chaotifying piecewise linear maps of the plane using a piecewise linear controller function is given. A domain of chaos in the resulting controlled map was determined exactly and rigorously.

The aim of this paper is to propose a variational piecewise constant level set method for solving elliptic shape and topology optimization problems. The original model is approximated by a two-phase optimal shape design problem by the ersatz material approach. Under the piecewise constant level set framework, we first reformulate the two-phase design problem to be a new constrained optimization...

This paper presents an H controller synthesis method for discrete time fuzzy dynamic systems based on a piecewise smooth Lyapunov function. The basic idea of the approach is to design a piecewise linear state feedback control law and use a piecewise smooth Lyapunov function to establish the global stability with H performance of the resulting closed loop fuzzy control systems. It is shown that ...

Piecewise affine systems are systems with a piecewise defined state equation. The continuous dynamics of these systems are affine but switch at the boundaries of certain subsets of the state-space. At these boundaries sliding modes may occur. A method will be presented to obtain a discrete-time system from a given continuous-time piecewise affine system. This method takes into account sliding m...

We define higher-order analogs to the piecewise linear surface finite element method studied in [Dz88] and prove error estimates in both pointwise and L2-based norms. Using the Laplace-Beltrami problem on an implicitly defined surface Γ as a model PDE, we define Lagrange finite element methods of arbitrary degree on polynomial approximations to Γ which likewise are of arbitrary degree. Then we ...

We introduce and analyse a finite volume method for the discretization of elliptic boundary value problems in R2. The method is based on nonuniform triangulations with piecewise linear nonconforming spaces. We prove optimal order error estimates in the L2–norm and a mesh dependent H1–norm.

This paper deals with the finite element approximation of the spectral problem for the Laplace equation with Neumann boundary conditions on a curved nonconvex domain Ω. Convergence and optimal order error estimates are proved for standard piecewise linear continuous elements on a discrete polygonal domain Ωh 6⊂ Ω in the framework of the abstract spectral approximation theory.

We consider standard finite volume piecewise linear approximations for second order elliptic boundary value problems on a nonconvex polygonal domain. Based on sharp shift estimates, we derive error estimations in H –, L2– and L∞–norm, taking into consideration the regularity of the data. Numerical experiments and counterexamples illustrate the theoretical results.

In this paper a piecewise linear finite element approximation of H-surfaces, or surfaces with constant mean curvature, spanned by a given Jordan curve in R3 is considered. It is proved that the finite element Hsurfaces converge to the exact H-surfaces under the condition that the Jordan curve is rectifiable. Several numerical examples are given.