1,762 research outputs found
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
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
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 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
Low Regularity Primal-Dual Weak Galerkin Finite Element Methods for Convection-Diffusion Equations
We propose a numerical method for convection-diffusion problems under low
regularity assumptions. We derive the method and analyze it using the
primal-dual weak Galerkin (PDWG) finite element framework. The Euler-Lagrange
formulation resulting from the PDWG scheme yields a system of equations
involving not only the equation for the primal variable but also its adjoint
for the dual variable. We show that the proposed PDWG method is stable and
convergent. We also derive a priori error estimates for the primal variable in
the -norm for . A series of numerical
tests that validate the theory and are presented as well.Comment: 28 pages, 5 figure
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
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
A Locking-Free Weak Galerkin Finite Element Method for Elasticity Problems in the Primal Formulation
This paper presents an arbitrary order locking-free numerical scheme for
linear elasticity on general polygonal/polyhedral partitions by using weak
Galerkin (WG) finite element methods. Like other WG methods, the key idea for
the linear elasticity is to introduce discrete weak strain and stress tensors
which are defined and computed by solving inexpensive local problems on each
element. Such local problems are derived from weak formulations of the
corresponding differential operators through integration by parts. Locking-free
error estimates of optimal order are derived in a discrete -norm and the
usual -norm for the approximate displacement when the exact solution is
smooth. Numerical results are presented to demonstrate the efficiency,
accuracy, and the locking-free property of the weak Galerkin finite element
method.Comment: 28 page
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
Superconvergence of Numerical Gradient for Weak Galerkin Finite Element Methods on Nonuniform Cartesian Partitions in Three Dimensions
A superconvergence error estimate for the gradient approximation of the
second order elliptic problem in three dimensions is analyzed by using weak
Galerkin finite element scheme on the uniform and non-uniform cubic partitions.
Due to the loss of the symmetric property from two dimensions to three
dimensions, this superconvergence result in three dimensions is not a trivial
extension of the recent superconvergence result in two dimensions
\cite{sup_LWW2018} from rectangular partitions to cubic partitions. The error
estimate for the numerical gradient in the -norm arrives at a
superconvergence order of when the lowest
order weak Galerkin finite elements consisting of piecewise linear polynomials
in the interior of the elements and piecewise constants on the faces of the
elements are employed. A series of numerical experiments are illustrated to
confirm the established superconvergence theory in three dimensions.Comment: 31 pages, 24 table
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