Energy norm error estimates for averaged discontinuous Galerkin methods: multidimensional case

A mathematical analysis is presented for a class of interior penalty (IP) discontinuous Galerkin approximations of elliptic boundary value problems. In the framework of the present theory one can derive some overpenalized IP bilinear forms in a natural way avoiding any heuristic choice of fluxes and penalty terms. The main idea is to start from bilinear forms for the local average of discontinuous approximations which are rewritten using the theory of distributions. It is pointed out that a class of overpenalized IP bilinear forms can be obtained using a lower order perturbation of these. Also, error estimations can be derived between the local averages of the discontinuous approximations and the analytic solution in the $H^1$-seminorm. Using the local averages, the analysis is performed in a conforming framework without any assumption on extra smoothness for the solution of the original boundary value problem

Fractional order elliptic problems with inhomogeneous Dirichlet boundary conditions

Fractional-order elliptic problems are investigated in case of inhomogeneous Dirichlet boundary data. The boundary integral form is proposed as a suitable mathematical model. The corresponding theory is completed by sharpening the mapping properties of the corresponding potential operators. Also a mild condition is provided to ensure the existence of the classical solution of the boundary integral equation

A new universal law for the Liesegang pattern formation

Classical regularities describing the Liesegang phenomenon have been observed and extensively studied in laboratory experiments for a long time. These have been verified in the last two decades, both theoretically and using simulations. However, they are only applicable if the observed system is driven by reaction and diffusion. We suggest here a new universal law, which is also valid in the case of various transport dynamics (purely diffusive, purely advective, and diffusion-advection cases). We state that ptot~Xc, where ptot yields the total amount of the precipitate and Xc is the center of gravity. Besides the theoretical derivation experimental and numerical evidence for the universal law is provided. In contrast to the classical regularities, the introduced quantities are continuous functions of time

A reliable and efficient implicit a posteriori error estimation technique for the time harmonic Maxwell equations

We analyze an implicit a posteriori error indicator for the time harmonic Maxwell equations and prove that it is both reliable and locally efficient. For the derivation, we generalize some recent results concerning explicit a posteriori error estimates. In particular, we relax the divergence free constraint for the source term. We also justify the complexity of the obtained estimator

Systematic front distortion and presence of consecutive fronts in a precipitation system

A new simple reaction-diffusion system is presented focusing on pattern formation phenomena as consecutive precipitation fronts and distortion of the precipitation front.The chemical system investigated here is based on the amphoteric property of aluminum hydroxide and exhibits two unique phenomena. Both the existence of consecutive precipitation fronts and distortion are reported for the first time. The precipitation patterns could be controlled by the pH field, and the distortion of the precipitation front can be practical for microtechnological applications of reaction-diffusion systems

Heterogeneous conductivity parameters in a one dimensional fuel cell model

A model of one dimensional fuel cells is investigated, where the material inhomogeneities in the cathode are taken into account. We use the results in some preceding studies to describe the dynamics of the chemical reactions and transport of ions. A corresponding governing equation is derived for the numerical simulations. We apply an explicit-implicit time integration and Richardson extrapolation technique to increase the accuracy of the approximations. The efficiency of the method is demonstrated using a non-trivial test problem with real parameters. Numerical simulations are executed in presence of inhomogeneous conductivities and their effect on the cell potential is investigated