1,428 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 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
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 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
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
Effective Implementation of the Weak Galerkin Finite Element Methods for the Biharmonic Equation
The weak Galerkin (WG) methods have been introduced in the references [11,
16] for solving the biharmonic equation. The purpose of this paper is to
develop an algorithm to implement the WG methods effectively. This can be
achieved by eliminating local unknowns to obtain a global system with
significant reduction of size. In fact, this reduced global system is
equivalent to the Schur complements of the WG methods. The unknowns of the
Schur complement of the WG method are those defined on the element boundaries.
The equivalence of the WG method and its Schur complement is established. The
numerical results demonstrate the effectiveness of this new implementation
technique.Comment: 10 page
A Hybridized Formulation for the Weak Galerkin Mixed Finite Element Method
This paper presents a hybridized formulation for the weak Galerkin mixed
finite element method (WG-MFEM) which was introduced and analyzed for second
order elliptic equations. The WG-MFEM method was designed by using
discontinuous piecewise polynomials on finite element partitions consisting of
polygonal or polyhedral elements of arbitrary shape. The key to WG-MFEM is the
use of a discrete weak divergence operator which is defined and computed by
solving inexpensive problems locally on each element. The hybridized
formulation of this paper leads to a significantly reduced system of linear
equations involving only the unknowns arising from the Lagrange multiplier in
hybridization. Optimal-order error estimates are derived for the hybridized
WG-MFEM approximations. Some numerical results are reported to confirm the
theory and a superconvergence for the Lagrange multiplier.Comment: 14 pages, 1 figure, 3 table
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
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