On the Penalty term for the Mixed Discontinuous Galerkin Finite Element Method for the Biharmonic Equation

In this paper, we present a study on the effect of penalty term in the mixed Discontinuous Galerkin Finite Element Method for the biharmonic equation proposed by \cite{gudi2008mixed}. The proposed mixed Discontinuous Galerkin Method showed sub-optimal rates of convergence for piecewise quadratic elements and no significant convergence rates for piecewise linear elements. We show that by choosing the penalty term proportional to $|e_k|^{-1}$ instead of $|e_k|^{-3}$, ensures an optimal rate of convergence for the approximation, including for piecewise linear elements. Finally, we present numerical experiments to validate our theoretical results.Comment: Replacement with major changes. Preprint versio

An Equilibration Based A Posteriori Error Estimate for the Biharmonic Equation and Two Finite Element Methods

We develop an a posteriori error estimator for the Interior Penalty Discontinuous Galerkin approximation of the biharmonic equation with continuous finite elements. The error bound is based on the two-energies principle and requires the computation of an equilibrated moment tensor. The natural space for the moment tensor consists of symmetric tensor fields with continuous normal-normal components. It is known from the Hellan-Herrmann-Johnson (HHJ) mixed formulation. We propose a construction that is totally local. The procedure can also be applied to the original HHJ formulation, which directly provides an equilibrated moment tensor

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 $H^2$ norm is established for the corresponding WG finite element solutions. Error estimates in the usual $L^2$ 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

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 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 $H^2$-equivalent norm for the WG finite element solutions. Error estimates in the usual $L^2$ 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

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 $H^2$-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 $H^2$-equivalent norm for the WG finite element solutions. Error estimates in the usual $L^2$ 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 C^0-Weak Galerkin Finite Element Method for the Biharmonic Equation

A C^0-weak Galerkin (WG) method is introduced and analyzed for solving the biharmonic equation in 2D and 3D. A weak Laplacian is defined for C^0 functions in the new weak formulation. This WG finite element formulation is symmetric, positive definite and parameter free. Optimal order error estimates are established in both a discrete H^2 norm and the L^2 norm, for the weak Galerkin finite element solution. Numerical results are presented to confirm the theory. As a technical tool, a refined Scott-Zhang interpolation operator is constructed to assist the corresponding error estimate. This refined interpolation preserves the volume mass of order (k+1-d) and the surface mass of order (k+2-d) for the P_{k+2} finite element functions in d-dimensional space.Comment: 21 page

The Weak Galerkin methods are rewritings of the Hybridizable Discontinuous Galerkin methods

We establish that the Weak Galerkin methods are rewritings of the hybridizable discontinuous Galerkin methods

A Weak Galerkin Finite Element Scheme for the Biharmonic Equations by Using Polynomials of Reduced Order

A new weak Galerkin (WG) finite element method for solving the biharmonic equation in two or three dimensional spaces by using polynomials of reduced order is introduced and analyzed. The WG method is on the use of weak functions and their weak derivatives defined as distributions. Weak functions and weak derivatives can be approximated by polynomials with various degrees. Different combination of polynomial spaces leads to different WG finite element methods, which makes WG methods highly flexible and efficient in practical computation. This paper explores the possibility of optimal combination of polynomial spaces that minimize the number of unknowns in the numerical scheme, yet without compromising the accuracy of the numerical approximation. Error estimates of optimal order are established for the corresponding WG approximations in both a discrete $H^2$ norm and the standard $L^2$ norm. In addition, the paper also presents some numerical experiments to demonstrate the power of the WG method. The numerical results show a great promise of the robustness, reliability, flexibility and accuracy of the WG method.Comment: 28 pages. arXiv admin note: substantial text overlap with arXiv:1303.0927, arXiv:1309.5560, arXiv:1510.06001 by other author

A Discontinuous Galerkin Method by Patch Reconstruction for Biharmonic Problem

We propose a new discontinuous Galerkin method based on the least-squares patch reconstruction for the biharmonic problem. We prove the optimal error estimate of the proposed method. The two-dimensional and three-dimensional numerical examples are presented to confirm the accuracy and efficiency of the method with several boundary conditions and several types of polygon meshes and polyhedral meshes