8,592 research outputs found
Variational formulation and numerical analysis of linear elliptic equations in nondivergence form with Cordes coefficients
This paper studies formulations of second-order elliptic partial differential
equations in nondivergence form on convex domains as equivalent variational
problems. The first formulation is that of Smears \& S\"uli [SIAM J.\ Numer.\
Anal.\ 51(2013), pp.\ 2088--2106.], and the second one is a new symmetric
formulation based on a least-squares functional. These formulations enable the
use of standard finite element techniques for variational problems in subspaces
of as well as mixed finite element methods from the context of fluid
computations. Besides the immediate quasi-optimal a~priori error bounds, the
variational setting allows for a~posteriori error control with explicit
constants and adaptive mesh-refinement. The convergence of an adaptive
algorithm is proved. Numerical results on uniform and adaptive meshes are
included
A mesh-free method for interface problems using the deep learning approach
In this paper, we propose a mesh-free method to solve interface problems
using the deep learning approach. Two interface problems are considered. The
first one is an elliptic PDE with a discontinuous and high-contrast
coefficient. While the second one is a linear elasticity equation with
discontinuous stress tensor. In both cases, we formulate the PDEs into
variational problems, which can be solved via the deep learning approach. To
deal with the inhomogeneous boundary conditions, we use a shallow neuron
network to approximate the boundary conditions. Instead of using an adaptive
mesh refinement method or specially designed basis functions or numerical
schemes to compute the PDE solutions, the proposed method has the advantages
that it is easy to implement and mesh-free. Finally, we present numerical
results to demonstrate the accuracy and efficiency of the proposed method for
interface problems
A Weak Galerkin Finite Element Method for A Type of Fourth Order Problem Arising From Fluorescence Tomography
In this paper, a new and efficient numerical algorithm by using weak Galerkin
(WG) finite element methods is proposed for a type of fourth order problem
arising from fluorescence tomography(FT). Fluorescence tomography is an
emerging, in vivo non-invasive 3-D imaging technique which reconstructs images
that characterize the distribution of molecules that are tagged by
fluorophores. Weak second order elliptic operator and its discrete version are
introduced for a class of discontinuous functions defined on a finite element
partition of the domain consisting of general polygons or polyhedra. An error
estimate of optimal order is derived in an -equivalent norm for the WG
finite element solutions. Error estimates in the usual norm are
established, yielding optimal order of convergence for all the WG finite
element algorithms except the one corresponding to the lowest order (i.e.,
piecewise quadratic elements). Some numerical experiments are presented to
illustrate the efficiency and accuracy of the numerical scheme.Comment: 27 pages,6 figures, 2 tables. arXiv admin note: substantial text
overlap with arXiv:1309.5560; substantial text overlap with arXiv:1303.0927
by other author
Non-negative mixed finite element formulations for a tensorial diffusion equation
We consider the tensorial diffusion equation, and address the discrete
maximum-minimum principle of mixed finite element formulations. In particular,
we address non-negative solutions (which is a special case of the
maximum-minimum principle) of mixed finite element formulations. The discrete
maximum-minimum principle is the discrete version of the maximum-minimum
principle.
In this paper we present two non-negative mixed finite element formulations
for tensorial diffusion equations based on constrained optimization techniques
(in particular, quadratic programming). These proposed mixed formulations
produce non-negative numerical solutions on arbitrary meshes for low-order
(i.e., linear, bilinear and trilinear) finite elements. The first formulation
is based on the Raviart-Thomas spaces, and is obtained by adding a non-negative
constraint to the variational statement of the Raviart-Thomas formulation. The
second non-negative formulation based on the variational multiscale
formulation.
For the former formulation we comment on the affect of adding the
non-negative constraint on the local mass balance property of the
Raviart-Thomas formulation. We also study the performance of the active set
strategy for solving the resulting constrained optimization problems. The
overall performance of the proposed formulation is illustrated on three
canonical test problems.Comment: 40 pages using amsart style file, and 15 figure
A scalable variational inequality approach for flow through porous media models with pressure-dependent viscosity
Mathematical models for flow through porous media typically enjoy the
so-called maximum principles, which place bounds on the pressure field. It is
highly desirable to preserve these bounds on the pressure field in predictive
numerical simulations, that is, one needs to satisfy discrete maximum
principles (DMP). Unfortunately, many of the existing formulations for flow
through porous media models do not satisfy DMP. This paper presents a robust,
scalable numerical formulation based on variational inequalities (VI), to model
non-linear flows through heterogeneous, anisotropic porous media without
violating DMP. VI is an optimization technique that places bounds on the
numerical solutions of partial differential equations. To crystallize the
ideas, a modification to Darcy equations by taking into account
pressure-dependent viscosity will be discretized using the lowest-order
Raviart-Thomas (RT0) and Variational Multi-scale (VMS) finite element
formulations. It will be shown that these formulations violate DMP, and, in
fact, these violations increase with an increase in anisotropy. It will be
shown that the proposed VI-based formulation provides a viable route to enforce
DMP. Moreover, it will be shown that the proposed formulation is scalable, and
can work with any numerical discretization and weak form. Parallel scalability
on modern computational platforms will be illustrated through strong-scaling
studies, which will prove the efficiency of the proposed formulation in a
parallel setting. Algorithmic scalability as the problem size is scaled up will
be demonstrated through novel static-scaling studies. The performed
static-scaling studies can serve as a guide for users to be able to select an
appropriate discretization for a given problem size
Gradient recovery for elliptic interface problem: I. body-fitted mesh
In this paper, we propose a novel gradient recovery method for elliptic
interface problem using body-fitted mesh in two dimension. Due to the lack of
regularity of solution at interface, standard gradient recovery methods fail to
give superconvergent results, and thus will lead to overrefinement when served
as a posteriori error estimator. This drawback is overcome by designing an
immersed gradient recovery operator in our method. We prove the
superconvergence of this method for both mildly unstructured mesh and adaptive
mesh, and present several numerical examples to verify the superconvergence and
its robustness as a posteriori error estimator
Vanishing moment method and moment solutions for second order fully nonlinear partial differential equations
This paper concerns with numerical approximations of solutions of second
order fully nonlinear partial differential equations (PDEs). A new notion of
weak solutions, called moment solutions, is introduced for second order fully
nonlinear PDEs. Unlike viscosity solutions, moment solutions are defined by a
constructive method, called vanishing moment method, hence, they can be readily
computed by existing numerical methods such as finite difference, finite
element, spectral Galerkin, and discontinuous Galerkin methods with
"guaranteed" convergence. The main idea of the proposed vanishing moment method
is to approximate a second order fully nonlinear PDE by a higher order, in
particular, a fourth order quasilinear PDE. We show by various numerical
experiments the viability of the proposed vanishing moment method. All our
numerical experiments show the convergence of the vanishing moment method, and
they also show that moment solutions coincide with viscosity solutions whenever
the latter exist.Comment: 24 pages and 30 figure
A Generalized Finite Element Method for the Obstacle Problem of Plates
A generalized finite element method for the displacement obstacle problem of
clamped Kirchhoff plates is considered in this paper. We derive optimal error
estimates and present numerical results that illustrate the performance of the
method
The Basics of Weak Galerkin Finite Element Methods
The goal of this article is to clarify some misunderstandings and
inappropriate claims made in [6] regarding the relation between the weak
Galerkin (WG) finite element method and the hybridizable discontinuous Galerkin
(HDG). In this paper, the authors offered their understandings and
interpretations on the weak Galerkin finite element method by describing the
basics of the WG method and how WG can be applied to a model PDE problem in
various variational forms. In the authors' view, WG-FEM and HDG methods are
based on different philosophies and therefore represent different methodologies
in numerical PDEs, though they share something in common in their roots. A
theory and an example are given to show that the primal WG-FEM is not
equivalent to the existing HDG [9]
A Preconditioned Descent Algorithm for Variational Inequalities of the Second Kind Involving the -Laplacian Operator
This paper is concerned with the numerical solution of a class of variational
inequalities of the second kind, involving the -Laplacian operator. This
kind of problems arise, for instance, in the mathematical modelling of
non-Newtonian fluids. We study these problems by using a regularization
approach, based on a Huber smoothing process. Well posedness of the regularized
problems is proved, and convergence of the regularized solutions to the
solution of the original problem is verified. We propose a preconditioned
descent method for the numerical solution of these problems and analyze the
convergence of this method in function spaces. The existence of admissible
descent directions is established by variational methods and admissible steps
are obtained by a backtracking algorithm which approximates the objective
functional by polynomial models. Finally, several numerical experiments are
carried out to show the efficiency of the methodology here introduced
- …