10,595 research outputs found
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 Weak Galerkin Mixed Finite Element Method for Second-Order Elliptic Problems
A new weak Galerkin (WG) method is introduced and analyzed for the second
order elliptic equation formulated as a system of two first order linear
equations. This method, called WG-MFEM, is designed by using discontinuous
piecewise polynomials on finite element partitions with arbitrary shape of
polygons/polyhedra. The WG-MFEM is capable of providing very accurate numerical
approximations for both the primary and flux variables. Allowing the use of
discontinuous approximating functions on arbitrary shape of polygons/polyhedra
makes the method highly flexible in practical computation. Optimal order error
estimates in both discrete and norms are established for the
corresponding weak Galerkin mixed finite element solutions.Comment: 26 page
A Weak Galerkin Finite Element Method for Second-Order Elliptic Problems
In this paper, authors shall introduce a finite element method by using a
weakly defined gradient operator over discontinuous functions with
heterogeneous properties. The use of weak gradients and their approximations
results in a new concept called {\em discrete weak gradients} which is expected
to play important roles in numerical methods for partial differential
equations. This article intends to provide a general framework for operating
differential operators on functions with heterogeneous properties. As a
demonstrative example, the discrete weak gradient operator is employed as a
building block to approximate the solution of a model second order elliptic
problem, in which the classical gradient operator is replaced by the discrete
weak gradient. The resulting numerical approximation is called a weak Galerkin
(WG) finite element solution. It can be seen that the weak Galerkin method
allows the use of totally discontinuous functions in the finite element
procedure. For the second order elliptic problem, an optimal order error
estimate in both a discrete and norms are established for the
corresponding weak Galerkin finite element solutions. A superconvergence is
also observed for the weak Galerkin approximation.Comment: 17 pages, research result
De Rham Complexes for Weak Galerkin Finite Element Spaces
Two de Rham complex sequences of the finite element spaces are introduced for
weak finite element functions and weak derivatives developed in the weak
Galerkin (WG) finite element methods on general polyhedral elements. One of the
sequences uses polynomials of equal order for all the finite element spaces
involved in the sequence and the other one uses polynomials of naturally
decending orders. It is shown that the diagrams in both de Rham complexes
commute for general polyhedral elements. The exactness of one of the complexes
is established for the lowest order element.Comment: 15 page
Interior penalty discontinuous Galerkin method on very general polygonal and polyhedral meshes
This paper focuses on interior penalty discontinuous Galerkin methods for
second order elliptic equations on very general polygonal or polyhedral meshes.
The mesh can be composed of any polygons or polyhedra which satisfies certain
shape regularity conditions characterized in a recent paper by two of the
authors in [17]. Such general meshes have important application in
computational sciences. The usual conforming finite element methods on
such meshes are either very complicated or impossible to implement in practical
computation. However, the interior penalty discontinuous Galerkin method
provides a simple and effective alternative approach which is efficient and
robust. This article provides a mathematical foundation for the use of interior
penalty discontinuous Galerkin methods in general meshes.Comment: 12 pages, research result
A Discrete Divergence-Free Weak Galerkin Finite Element Method for the Stokes Equations
A discrete divergence-free weak Galerkin finite element method is developed
for the Stokes equations based on a weak Galerkin (WG) method introduced in the
reference [15]. Discrete divergence-free bases are constructed explicitly for
the lowest order weak Galerkin elements in two and three dimensional spaces.
These basis functions can be derived on general meshes of arbitrary shape of
polygons and polyhedrons. With the divergence-free basis derived, the discrete
divergence-free WG scheme can eliminate the pressure variable from the system
and reduces a saddle point problem to a symmetric and positive definite system
with many fewer unknowns. Numerical results are presented to demonstrate the
robustness and accuracy of this discrete divergence-free WG method.Comment: 12 page
Newton Method for Sparse Logistic Regression: Quadratic Convergence and Extensive Simulations
Sparse logistic regression, {as an effective tool of classification,} has
been developed tremendously in recent two decades, from its origination the
-regularized version to the sparsity constrained models. This paper is
carried out on the sparsity constrained logistic regression by the Newton
method. We begin with establishing its first-order optimality condition
associated with a -stationary point. This point can be equivalently
interpreted as an equation system which is then efficiently solved by the
Newton method. The method has a considerably low computational complexity and
enjoys global and quadratic convergence properties. Numerical experiments on
random and real data demonstrate its superior performance when against seven
state-of-the-art solvers
A Stable Numerical Algorithm for the Brinkman Equations by Weak Galerkin Finite Element Methods
This paper presents a stable numerical algorithm for the Brinkman equations
by using weak Galerkin (WG) finite element methods. The Brinkman equations can
be viewed mathematically as a combination of the Stokes and Darcy equations
which model fluid flow in a multi-physics environment, such as flow in complex
porous media with a permeability coefficient highly varying in the simulation
domain. In such applications, the flow is dominated by Darcy in some regions
and by Stokes in others. It is well known that the usual Stokes stable elements
do not work well for Darcy flow and vise versa. The challenge of this study is
on the design of numerical schemes which are stable for both the Stokes and the
Darcy equations. This paper shows that the WG finite element method is capable
of meeting this challenge by providing a numerical scheme that is stable and
accurate for both Darcy and the Stokes dominated flows. Error estimates of
optimal order are established for the corresponding WG finite element
solutions. The paper also presents some numerical experiments that demonstrate
the robustness, reliability, flexibility and accuracy of the WG method for the
Brinkman equations.Comment: 20 pages, 21 plots and figure
Weak Galerkin Finite Element Methods for the Biharmonic Equation on Polytopal Meshes
A new weak Galerkin (WG) finite element method is introduced and analyzed in
this paper for the biharmonic equation in its primary form. This method is
highly robust and flexible in the element construction by using discontinuous
piecewise polynomials on general finite element partitions consisting of
polygons or polyhedra of arbitrary shape. The resulting WG finite element
formulation is symmetric, positive definite, and parameter-free. Optimal order
error estimates in a discrete norm is established for the corresponding
WG finite element solutions. Error estimates in the usual norm are also
derived, yielding a sub-optimal order of convergence for the lowest order
element and an optimal order of convergence for all high order of elements.
Numerical results are presented to confirm the theory of convergence under
suitable regularity assumptions.Comment: 23 pages, 1 figure, 2 tables. arXiv admin note: text overlap with
arXiv:1202.3655, arXiv:1204.365
Weak Galerkin Finite Element Methods on Polytopal Meshes
This paper introduces a new weak Galerkin (WG) finite element method for
second order elliptic equations on polytopal meshes. This method, called
WG-FEM, is designed by using a discrete weak gradient operator applied to
discontinuous piecewise polynomials on finite element partitions of arbitrary
polytopes with certain shape regularity. The paper explains how the numerical
schemes are designed and why they provide reliable numerical approximations for
the underlying partial differential equations. In particular, optimal order
error estimates are established for the corresponding WG-FEM approximations in
both a discrete norm and the standard norm. Numerical results are
presented to demonstrate the robustness, reliability, and accuracy of the
WG-FEM. All the results are derived for finite element partitions with
polytopes. Allowing the use of discontinuous approximating functions on
arbitrary polytopal elements is a highly demanded feature for numerical
algorithms in scientific computing.Comment: 22 pages, 4 figures, 5 table
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