2,462 research outputs found
Adaptive BEM with optimal convergence rates for the Helmholtz equation
We analyze an adaptive boundary element method for the weakly-singular and
hypersingular integral equations for the 2D and 3D Helmholtz problem. The
proposed adaptive algorithm is steered by a residual error estimator and does
not rely on any a priori information that the underlying meshes are
sufficiently fine. We prove convergence of the error estimator with optimal
algebraic rates, independently of the (coarse) initial mesh. As a technical
contribution, we prove certain local inverse-type estimates for the boundary
integral operators associated with the Helmholtz equation
A Generalized Multiscale Finite Element Method for the Brinkman Equation
In this paper we consider the numerical upscaling of the Brinkman equation in
the presence of high-contrast permeability fields. We develop and analyze a
robust and efficient Generalized Multiscale Finite Element Method (GMsFEM) for
the Brinkman model. In the fine grid, we use mixed finite element method with
the velocity and pressure being continuous piecewise quadratic and piecewise
constant finite element spaces, respectively. Using the GMsFEM framework we
construct suitable coarse-scale spaces for the velocity and pressure that yield
a robust mixed GMsFEM. We develop a novel approach to construct a coarse
approximation for the velocity snapshot space and a robust small offline space
for the velocity space. The stability of the mixed GMsFEM and a priori error
estimates are derived. A variety of two-dimensional numerical examples are
presented to illustrate the effectiveness of the algorithm.Comment: 22 page
New Results on Triangulation, Polynomial Equation Solving and Their Application in Global Localization
This thesis addresses the problem of global localization from images. The overall goal is to find the location and the direction of a camera given an image taken with the camera relative a 3D world model. In order to solve the problem several subproblems have to be handled. The two main steps for constructing a system for global localization consist of model building and localization. For the model construction phase we give a new method for triangulation that guarantees that the globally optimal position is attained under the assumption of Gaussian noise in the image measurements. A common framework for the triangulation of points, lines and conics is presented. The second contribution of the thesis is in the field of solving systems of polynomial equations. Many problems in geometrical computer vision lead to computing the real roots of a system of polynomial equations, and several such geometry problems appear in the localization problem. The method presented in the thesis gives a significant improvement in the numerics when Gröbner basis methods are applied. Such methods are often plagued by numerical problems, but by using the fact that the complete Gröbner basis is not needed, the numerics can be improved. In the final part of the thesis we present several new minimal, geometric problems that have not been solved previously. These minimal cases make use of both two and three dimensional correspondences at the same time. The solutions to these minimal problems form the basis of a localization system which aims at improving robustness compared to the state of the art
Trace Finite Element Methods for PDEs on Surfaces
In this paper we consider a class of unfitted finite element methods for
discretization of partial differential equations on surfaces. In this class of
methods known as the Trace Finite Element Method (TraceFEM), restrictions or
traces of background surface-independent finite element functions are used to
approximate the solution of a PDE on a surface. We treat equations on steady
and time-dependent (evolving) surfaces. Higher order TraceFEM is explained in
detail. We review the error analysis and algebraic properties of the method.
The paper navigates through the known variants of the TraceFEM and the
literature on the subject
Incompressible Lagrangian fluid flow with thermal coupling
In this monograph is presented a method for the solution of an incompressible viscous fluid flow with heat transfer and solidification usin a fully Lagrangian description on the motion. The originality of this method consists in assembling various concepts and techniques which appear naturally due to the Lagrangian formulation.Postprint (published version
An adaptive octree finite element method for PDEs posed on surfaces
The paper develops a finite element method for partial differential equations
posed on hypersurfaces in , . The method uses traces of
bulk finite element functions on a surface embedded in a volumetric domain. The
bulk finite element space is defined on an octree grid which is locally refined
or coarsened depending on error indicators and estimated values of the surface
curvatures. The cartesian structure of the bulk mesh leads to easy and
efficient adaptation process, while the trace finite element method makes
fitting the mesh to the surface unnecessary. The number of degrees of freedom
involved in computations is consistent with the two-dimension nature of surface
PDEs. No parametrization of the surface is required; it can be given implicitly
by a level set function. In practice, a variant of the marching cubes method is
used to recover the surface with the second order accuracy. We prove the
optimal order of accuracy for the trace finite element method in and
surface norms for a problem with smooth solution and quasi-uniform mesh
refinement. Experiments with less regular problems demonstrate optimal
convergence with respect to the number of degrees of freedom, if grid
adaptation is based on an appropriate error indicator. The paper shows results
of numerical experiments for a variety of geometries and problems, including
advection-diffusion equations on surfaces. Analysis and numerical results of
the paper suggest that combination of cartesian adaptive meshes and the
unfitted (trace) finite elements provide simple, efficient, and reliable tool
for numerical treatment of PDEs posed on surfaces
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