Navigation is a major challenge for autonomous, mobile robots. The problem can basically be divided into positioning and path planning. In this paper we present an approach which we call grid-based navigation. Though we also propose a scheme for path finding, we focus on positioning. Our approach uses minimal environmental infrastructure and only two light sensors on the mobile device itself. S...

We propose a generalized multiscale finite element method (GMsFEM) based on clustering algorithm to study the elliptic PDEs with random coefficients in the multiquery setting. Our method consists of offline and online stages. In the offline stage, we construct a small number of reduced basis functions within each coarse grid block, which can then be used to approximate the multiscale finite ele...

Modeling of wave propagation in a heterogeneous medium requires input data that varies on many different spatial and temporal scales. Operator-based upscaling allows us to capture the effect of the fine scales on a coarser domain without solving the full fine-scale problem. The method applied to the constant density, variable sound velocity acoustic wave equation consists of two stages. First, ...

Conservation properties of the mass, momentum, and kinetic energy equations for incompressible flow are specified as analytical requirements for a proper set of discrete equations. Existing finite difference schemes in regular and staggered grid systems are checked for violations of the conservation requirements and a few important discrepancies are pointed out. In particular, it is found that ...

Standard spectral methods are capable of providing very accurate approximations to well-behaved smooth functions with significantly less degrees of freedom when compared with finite difference or finite element methods (cf. [6,7,11]). However, if a function exhibits localized behaviors such as sharp interfaces, very thin internal or boundary layers, using a standard Gauss-type grid usually fail...

In this thesis we study so-called two-grid hp-version discontinuous Galerkin finite element methods for the numerical solution of quasilinear partial differential equations. The two-grid method is constructed by first solving the nonlinear system of equations stemming from the discontinuous Galerkin finite element method on a coarse mesh partition; then, this coarse solution is used to linearis...

Themiscible displacement of one incompressible fluid by another in a porous medium is governed by a system of two equations. One is elliptic form equation for the pressure and the other is parabolic form equation for the concentration of one of the fluids. Since only the velocity and not the pressure appears explicitly in the concentration equation, we use a mixed finite element method for the ...

Adaptive mesh refinement techniques offer a flexible framework for future variable-resolution climate and weather models since they can focus their computational mesh on certain geographical areas or atmospheric events. Adaptive meshes can also be used to coarsen a latitude-longitude grid in polar regions. This allows for the so-called reduced grid setups. A spherical, block-structured adaptive...

The results of an investigation in developing a generalized toolkit for Quadtree based Finite Volume models, exemplarily applied on the Shallow Water Equations, are given in this report. The developed application model: (i) discretizes a given topography using a quadtreebased grid system and (ii) solves the Shallow Water Equations by a numerical method based on the Finite Volume Method on the a...