47,368 research outputs found
Adaptive Pseudo-Transient-Continuation-Galerkin Methods for Semilinear Elliptic Partial Differential Equations
In this paper we investigate the application of pseudo-transient-continuation
(PTC) schemes for the numerical solution of semilinear elliptic partial
differential equations, with possible singular perturbations. We will outline a
residual reduction analysis within the framework of general Hilbert spaces,
and, subsequently, employ the PTC-methodology in the context of finite element
discretizations of semilinear boundary value problems. Our approach combines
both a prediction-type PTC-method (for infinite dimensional problems) and an
adaptive finite element discretization (based on a robust a posteriori residual
analysis), thereby leading to a fully adaptive PTC-Galerkin scheme. Numerical
experiments underline the robustness and reliability of the proposed approach
for different examples.Comment: arXiv admin note: text overlap with arXiv:1408.522
Multiscale Finite Element Methods for Nonlinear Problems and their Applications
In this paper we propose a generalization of multiscale finite element methods (Ms-FEM) to nonlinear problems. We study the convergence of the proposed method for nonlinear elliptic equations and propose an oversampling technique. Numerical examples demonstrate that the over-sampling technique greatly reduces the error. The application of MsFEM to porous media flows is considered. Finally, we describe further generalizations of MsFEM to nonlinear time-dependent equations and discuss the convergence of the method for various kinds of heterogeneities
Optimization of the Penalty Parameter for the Dual-Wind Discontinuous Galerkin Methods on a Prototypical Second Order Pde.
A discontinuous Galerkin (DG) finite-element interior calculus is used as a common framework to describe various DG approximation methods for second- order elliptic problems. The dual-wind discontinuous Galerkin method (DWDG) has been shown to be stable and consistent for a wide range of penalty parameter values, including zero, for second order elliptic problems under certain mesh conditions. In this presentation, we will present the results of numerical experiments on various second order elliptic problems with varying penalty parameters that show the choice of zero for the penalty parameter is an optimal choice for application
Finite element solution for elliptic partial differential equations
The contents of this thesis are a detailed study of the implementation
of Finite Element method for solving linear and non-linear elliptic
partial differential equations. It commences with a description and
classification of partial differential equations, the related matrix and
eigenvalue theory and the related matrix methods to solve the linear and
non-linear systems of equations.
In Chapter Three, we discuss the development of the, finite element
method and its application with a full description of an orderly step-by-step
process. In Chapter Four, we discuss the implementation of developing
an efficient easy-to-use finite element program for the general two-dimensional
problem along with the capability of handling problems for
different domains and boundary conditions and with a fully automated mesh
generation and refinement technique along with a description of generalised
pre- and post-processors for the Finite Element Method. [Continues.
hp-Version Composite Discontinuous Galerkin Methods for Elliptic Problems on Complicated Domains
In this paper we introduce the hp-version discontinuous Galerkin composite finite element method for the discretization of second-order elliptic partial differential equations. This class of methods allows for the approximation of problems posed on computational domains which may contain a huge number of local geometrical features, or microstructures. While standard numerical methods can be devised for such problems, the computational effort may be extremely high, as the minimal number of elements needed to represent the underlying domain can be very large. In contrast, the minimal dimension of the underlying composite finite element space is independent of the number of geometric features. The key idea in the construction of this latter class of methods is that the computational domain Ω is no longer resolved by the mesh; instead, the finite element basis (or shape) functions are adapted to the geometric details present in Ω. In this paper, we extend these ideas to the discontinuous Galerkin setting, based on employing the hp-version of the finite element method. Numerical experiments highlighting the practical application of the proposed numerical scheme will be presented
Error estimates for finite element approximations of nonlinear monotone elliptic problems with application to numerical homogenization
We consider a finite element method (FEM) with arbitrary polynomial degree for nonlinear monotone elliptic problems. Using a linear elliptic projection, we first give a new short proof of the optimal convergence rate of the FEM in the L2 norm. We then derive optimal a priori error estimates in the H 1 and L2 norm for a FEM with variational crimes due to numerical integration. As an application we derive a priori error estimates for a numerical homogenization method applied to nonlinear monotone elliptic problems
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