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

Models of Liesegang pattern formation

In this article different mathematical models of the Liesegang phenomenon are exhibited. The main principles of modeling are discussed such as supersaturation theory, sol coagulation and phase separation, which describe the phenomenon using different steps and mechanism beyond the simple reaction scheme. We discuss whether the underlying numerical simulations are able to reproduce several empirical regularities and laws of the corresponding pattern structure. In all cases we highlight the meaning of the initial and boundary conditions in the corresponding mathematical formalism. Above the deterministic ones discrete stochastic approaches are also described. As a main tool for the control of pattern structure the effect of an external electric field is also discussed

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

Models of space-fractional diffusion: a critical review

Space-fractional diffusion problems are investigated from the modeling point of view. It is pointed out that the elementwise power of the Laplacian operator in R n is an inadequate model of fractional diffusion. Also, the approach with fractional calculus using zero extension is not a proper model of homogeneous Dirichlet boundary conditions. At the time, the spectral definition of the fractional Dirichlet Laplacian seems to be in many aspects a proper model of fractional diffusion

CONVERGENCE OF THE MATRIX TRANSFORMATION METHOD FOR THE FINITE DIFFERENCE APPROXIMATION OF FRACTIONAL ORDER DIFFUSION PROBLEMS

Numerical solution of fractional order diffusion problems with homogeneous Dirichlet boundary conditions is investigated on a square domain. An appropriate extension is applied to have a well-posed problem on R 2 and the solution on the square is regarded as a localization. For the numerical approximation a finite difference method is applied combined with the matrix transformation method. Here the discrete fractional Laplacian is approximated with a matrix power instead of computing the complicated approxima- tions of fractional order derivatives. The spatial convergence of this method is proved and demonstrated in some numerical experiments

Stability of reaction fronts in random walk simulations

A model of propagating reaction fronts is given for simple autocatalytic reactions and the stability of the propagating reaction fronts are studied in several numerical experiments. The corresponding random walk simulations - extending of a recent algorithm - make possible the simultaneous treatment of moving particles. A systematic comparison with the standard deterministic simulations highlight the advantages of the present stochastic approach. The main favor of the random walk simulation is that the initial perturbation has no strong effect on the stability of the front unlike in deterministic cases

Energy norm error estimates for averaged discontinuous Galerkin methods in 1 dimension

Numerical solution of one-dimensional elliptic problems is investigated using an averaged discontinuous discretization. The corresponding numerical method can be performed using the favorable properties of the discontinuous Galerkin (dG) approach, while for the average an error estimation is obtained in the i?1-seminorm. We point out that this average can be regarded as a lower order modification of the average of a well-known overpenalized symmetric interior penalty (IP) method. This allows a natural derivation of the overpenalized IP methods. © 2014 Institute for Scientific Computing and Information