1,003 research outputs found
Discretization of div-curl Systems by Weak Galerkin Finite Element Methods on Polyhedral Partitions
In this paper, the authors devise a new discretization scheme for div-curl
systems defined in connected domains with heterogeneous media by using the weak
Galerkin finite element method. Two types of boundary value problems are
considered in the algorithm development: (1) normal boundary condition, and (2)
tangential boundary condition. A new variational formulation is developed for
the normal boundary value problem by using the Helmholtz decomposition which
avoids the computation of functions in the harmonic fields. Both boundary value
problems are reduced to a general saddle-point problem involving the curl and
divergence operators, for which the weak Galerkin finite element method is
devised and analyzed. The novelty of the technique lies in the discretization
of the divergence operator applied to vector fields with heterogeneous media.
Error estimates of optimal order are established for the corresponding finite
element approximations in various discrete Sobolev norms.Comment: 27 page
An Efficient Numerical Scheme for the Biharmonic Equation by Weak Galerkin Finite Element Methods on Polygonal or Polyhedral Meshes
This paper presents a new and efficient numerical algorithm for the
biharmonic equation by using weak Galerkin (WG) finite element methods. The WG
finite element scheme is based on a variational form of the biharmonic equation
that is equivalent to the usual -semi norm. Weak partial derivatives and
their approximations, called discrete weak partial derivatives, are introduced
for a class of discontinuous functions defined on a finite element partition of
the domain consisting of general polygons or polyhedra. The discrete weak
partial derivatives serve as building blocks for the WG finite element method.
The resulting matrix from the WG method is symmetric, positive definite, and
parameter free. 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: 25 pages. arXiv admin note: text overlap with arXiv:1303.092
A Primal-Dual Weak Galerkin Finite Element Method for Second Order Elliptic Equations in Non-Divergence Form
This article proposes a new numerical algorithm for second order elliptic
equations in non-divergence form. The new method is based on a discrete weak
Hessian operator locally constructed by following the weak Galerkin strategy.
The numerical solution is characterized as a minimization of a non-negative
quadratic functional with constraints that mimic the second order elliptic
equation by using the discrete weak Hessian. The resulting Euler-Lagrange
equation offers a symmetric finite element scheme involving both the primal and
a dual variable known as the Lagrange multiplier, and thus the name of
primal-dual weak Galerkin finite element method. Optimal order error estimates
are derived for the finite element approximations in a discrete -norm, as
well as the usual - and -norms. Some numerical results are presented
for smooth and non-smooth coefficients on convex and non-convex domains.Comment: 30 pages, 10 table
A Primal-Dual Weak Galerkin Finite Element Method for Fokker-Planck Type Equations
This paper presents a primal-dual weak Galerkin (PD-WG) finite element method
for a class of second order elliptic equations of Fokker-Planck type. The
method is based on a variational form where all the derivatives are applied to
the test functions so that no regularity is necessary for the exact solution of
the model equation. The numerical scheme is designed by using locally
constructed weak second order partial derivatives and the weak gradient
commonly used in the weak Galerkin context. Optimal order of convergence is
derived for the resulting numerical solutions. Numerical results are reported
to demonstrate the performance of the numerical scheme.Comment: 26 pages, 10 tables, 3 figure
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
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 discrete maximum principle for the weak Galerkin finite element method on nonuniform rectangular partitions
This article establishes a discrete maximum principle (DMP) for the
approximate solution of convection-diffusion-reaction problems obtained from
the weak Galerkin finite element method on nonuniform rectangular partitions.
The DMP analysis is based on a simplified formulation of the weak Galerkin
involving only the approximating functions defined on the boundary of each
element. The simplified weak Galerkin method has a reduced computational
complexity over the usual weak Galerkin, and indeed provides a discretization
scheme different from the weak Galerkin when the reaction term presents. An
application of the simplified weak Galerkin on uniform rectangular partitions
yields some - and -point finite difference schemes for the second order
elliptic equation. Numerical experiments are presented to verify the discrete
maximum principle and the accuracy of the scheme, particularly the finite
difference scheme
Simplified Weak Galerkin and New Finite Difference Schemes for the Stokes Equation
This article presents a simplified formulation for the weak Galerkin finite
element method for the Stokes equation without using the degrees of freedom
associated with the unknowns in the interior of each element as formulated in
the original weak Galerkin algorithm. The simplified formulation preserves the
important mass conservation property locally on each element and allows the use
of general polygonal partitions. A particular application of the simplified
weak Galerkin on rectangular partitions yields a new class of 5- and 7-point
finite difference schemes for the Stokes equation. An explicit formula is
presented for the computation of the element stiffness matrices on arbitrary
polygonal elements. Error estimates of optimal order are established for the
simplified weak Galerkin finite element method in the H^1 and L^2 norms.
Furthermore, a superconvergence of order O(h^{1.5}) is established on
rectangular partitions for the velocity approximation in the H^1 norm at cell
centers, and a similar superconvergence is derived for the pressure
approximation in the L^2 norm at cell centers. Some numerical results are
reported to confirm the convergence and superconvergence theory.Comment: 32 pages, 7 figures, 2 table
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
Superconvergence of the Gradient Approximation for Weak Galerkin Finite Element Methods on Nonuniform Rectangular Partitions
This article presents a superconvergence for the gradient approximation of
the second order elliptic equation discretized by the weak Galerkin finite
element methods on nonuniform rectangular partitions. The result shows a
convergence of , , for the numerical gradient
obtained from the lowest order weak Galerkin element consisting of piecewise
linear and constant functions. For this numerical scheme, the optimal order of
error estimate is for the gradient approximation. The
superconvergence reveals a superior performance of the weak Galerkin finite
element methods. Some computational results are included to numerically
validate the superconvergence theory
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