79 research outputs found
A simplified primal-dual weak Galerkin finite element method for Fokker-Planck type equations
A simplified primal-dual weak Galerkin (S-PDWG) finite element method is
designed for the Fokker-Planck type equation with non-smooth diffusion tensor
and drift vector. The discrete system resulting from S-PDWG method has
significantly fewer degrees of freedom compared with the one resulting from the
PDWG method proposed by Wang-Wang \cite{WW-fp-2018}. Furthermore, the condition
number of the S-PDWG method is smaller than the PDWG method \cite{WW-fp-2018}
due to the introduction of a new stabilizer, which provides a potential for
designing fast algorithms. Optimal order error estimates for the S-PDWG
approximation are established in the norm. A series of numerical results
are demonstrated to validate the effectiveness of the S-PDWG method.Comment: 23 pages, 17 table
A Unified Study of Continuous and Discontinuous Galerkin Methods
A unified study is presented in this paper for the design and analysis of
different finite element methods (FEMs), including conforming and nonconforming
FEMs, mixed FEMs, hybrid FEMs,discontinuous Galerkin (DG) methods, hybrid
discontinuous Galerkin (HDG) methods and weak Galerkin (WG) methods. Both HDG
and WG are shown to admit inf-sup conditions that hold uniformly with respect
to both mesh and penalization parameters. In addition, by taking the limit of
the stabilization parameters, a WG method is shown to converge to a mixed
method whereas an HDG method is shown to converge to a primal method.
Furthermore, a special class of DG methods, known as the mixed DG methods, is
presented to fill a gap revealed in the unified framework.Comment: 39 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
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
Low Regularity Primal-Dual Weak Galerkin Finite Element Methods for Ill-Posed Elliptic Cauchy Problems
A new primal-dual weak Galerkin (PDWG) finite element method is introduced
and analyzed for the ill-posed elliptic Cauchy problems with ultra-low
regularity assumptions on the exact solution. The Euler-Lagrange formulation
resulting from the PDWG scheme yields a system of equations involving both the
primal equation and the adjoint (dual) equation. The optimal order error
estimate for the primal variable in a low regularity assumption is established.
A series of numerical experiments are illustrated to validate effectiveness of
the developed theory.Comment: 20 pages, 18 table
Bregman Distances in Inverse Problems and Partial Differential Equation
The aim of this paper is to provide an overview of recent development related
to Bregman distances outside its native areas of optimization and statistics.
We discuss approaches in inverse problems and image processing based on Bregman
distances, which have evolved to a standard tool in these fields in the last
decade. Moreover, we discuss related issues in the analysis and numerical
analysis of nonlinear partial differential equations with a variational
structure. For such problems Bregman distances appear to be of similar
importance, but are currently used only in a quite hidden fashion. We try to
work out explicitely the aspects related to Bregman distances, which also lead
to novel mathematical questions and may also stimulate further research in
these areas
Primal-Dual Weak Galerkin Finite Element Methods for Elliptic Cauchy Problems
The authors propose and analyze a well-posed numerical scheme for a type of
ill-posed elliptic Cauchy problem by using a constrained minimization approach
combined with the weak Galerkin finite element method. The resulting
Euler-Lagrange formulation yields a system of equations involving the original
equation for the primal variable and its adjoint for the dual variable, and is
thus an example of the primal-dual weak Galerkin finite element method. This
new primal-dual weak Galerkin algorithm is consistent in the sense that the
system is symmetric, well-posed, and is satisfied by the exact solution. A
certain stability and error estimates were derived in discrete Sobolev norms,
including one in a weak topology. Some numerical results are reported to
illustrate and validate the theory developed in the paper.Comment: 27 pages, 4 figure
New Primal-Dual Weak Galerkin Finite Element Methods for Convection-Diffusion Problems
This article devises a new primal-dual weak Galerkin finite element method
for the convection-diffusion equation. Optimal order error estimates are
established for the primal-dual weak Galerkin approximations in various
discrete norms and the standard norms. A series of numerical experiments
are conducted and reported to verify the theoretical findings.Comment: 26 pages, 3 figures, 9 table
Primal dual mixed finite element methods for the elliptic Cauchy problem
We consider primal-dual mixed finite element methods for the solution of the
elliptic Cauchy problem, or other related data assimilation problems. The
method has a local conservation property. We derive a priori error estimates
using known conditional stability estimates and determine the minimal amount of
weakly consistent stabilization and Tikhonov regularization that yields optimal
convergence for smooth exact solutions. The effect of perturbations in data is
also accounted for. A reduced version of the method, obtained by choosing a
special stabilization of the dual variable, can be viewed as a variant of the
least squares mixed finite element method introduced by Dard\'e, Hannukainen
and Hyv\"onen in \emph{An {H\sb {\sf{div}}}-based mixed quasi-reversibility
method for solving elliptic {C}auchy problems}, SIAM J. Numer. Anal., 51(4)
2013. The main difference is that our choice of regularization does not depend
on auxiliary parameters, the mesh size being the only asymptotic parameter.
Finally, we show that the reduced method can be used for defect correction
iteration to determine the solution of the full method. The theory is
illustrated by some numerical examples
A Locking-Free Finite Element Method for Linear Elasticity Equations on Polytopal Partitions
This article presents a finite element method for boundary value
problems for linear elasticity equations. The new method makes use of piecewise
constant approximating functions on the boundary of each polytopal element, and
is devised by simplifying and modifying the weak Galerkin finite element method
based on approximations for the displacement. This new scheme
includes a tangential stability term on top of the simplified weak Galerkin to
ensure the necessary stability due to the rigid motion. The new method involves
a small number of unknowns on each element; it is user-friendly in computer
implementation; and the element stiffness matrix can be easily computed for
general polytopal elements. The numerical method is of second order accurate,
locking-free in the nearly incompressible limit, ease polytopal partitions in
practical computation. Error estimates in , , and some negative norms
are established for the corresponding numerical displacement. Numerical results
are reported for several 2D and 3D test problems, including the classical
benchmark Cook's membrane problem in two dimensions as well as some three
dimensional problems involving shear loaded phenomenon. The numerical results
show clearly the simplicity, stability, accuracy, and the efficiency of the new
method
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