A modified weak Galerkin method for the biharmonic equation

Effective implementation of the weak Galerkin finite element methods for the biharmonic equation

A \(C^0\)-weak Galerkin (WG) method is introduced and analyzed in this article for solving the biharmonic equation in 2D and 3D. A discrete weak Laplacian is defined for \(C^0\) functions, which is then used to design the weak Galerkin finite element scheme. This WG finite element formulation is symmetric, positive definite and parameter free. Optimal order error estimates are established for the weak Galerkin finite element solution in both a discrete \(H^2\) norm and the standard \(H^1\) and \(L^2\) norms with appropriate regularity assumptions. 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 estimates. 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.

The energy-diminishing weak Galerkin finite element method for the computation of ground state and excited states in Bose-Einstein condensates

Error analysis of weak Galerkin finite element methods for time-dependent convection–diffusion equations

Weak Galerkin methods for second order elliptic interface problems

In this paper, we investigate a two‐grid weak Galerkin method for semilinear elliptic differential equations. The method mainly contains two steps. First, we solve the semilinear elliptic equation on the coarse mesh with mesh size , then, we use the coarse mesh solution as an initial guess to linearize the semilinear equation on the fine mesh, that is, on the fine mesh (with mesh size ), we only need to solve a linearized system. Theoretical analysis shows that when the exact solution has sufficient regularity and , the two‐grid weak Galerkin method achieves the same convergence accuracy as weak Galerkin method. Several examples are given to verify the theoretical results.

A new finite element method is developed for the Reissner–Mindlin equations in its primary form by using the weak Galerkin approach. Like other weak Galerkin finite element methods, this one is highly flexible and robust by allowing the use of discontinuous approximating functions on arbitrary shape of polygons and, at the same time, is parameter independent on its stability and convergence. Error estimates of optimal order in mesh size h are established for the corresponding weak Galerkin approximations. Numerical experiments are conducted for verifying the convergence theory, as well as suggesting some superconvergence and a uniform convergence of the method with respect to the plate thickness.

<p style='text-indent:20px;'>The weak Galerkin (WG) finite element method is an effective and flexible general numerical technique for solving partial differential equations. It is a natural extension of the classic conforming finite element method for discontinuous approximations, which maintains simple finite element formulation. Stabilizer free weak Galerkin methods further simplify the WG methods and reduce computational complexity. This paper explores the possibility of optimal combination of polynomial spaces that minimize the number of unknowns in the stabilizer free WG schemes without compromising the accuracy of the numerical approximation. A new stabilizer free weak Galerkin finite element method is proposed and analyzed with polynomial degree reduction. To achieve such a goal, a new definition of weak gradient is introduced. Error estimates of optimal order are established for the corresponding WG approximations in both a discrete <inline-formula><tex-math id="M1">\begin{document}$ H^1 $\end{document}</tex-math></inline-formula> norm and the standard <inline-formula><tex-math id="M2">\begin{document}$ L^2 $\end{document}</tex-math></inline-formula> norm. The numerical examples are tested on various meshes and confirm the theory.

A potential-robust WG (weak Galerkin) method is introduced for the Maxwell equations. We obtain potential/magnetic-permeability independent error estimates. Optimal-order convergence rates are proved in both the energy norm and the L 2 {L^{2}} norm. Numerical examples verify the theory.

Brute force three dimensional simulation of Surface Acoustic Wave (SAW) and Bulk Acoustic Wave (BAW) devices with massive metallic electrodes of finite-aperture, massive busbars and massive protective closure structures is a prohibitively difficult task. This paper presents conceptually new ideas and implementation tools based on a mesh-less analysis of the problem, to tackle the daunting task of modelling 3D massloading effect with high accuracy. A sophisticated combination of spectral theory, weak Galerkin method and Green's function theory has been employed. A proof of concept has been provided.

Complete discrete error analysis of implicit Newmark-(γ, θ)-schemes for the acoustic wave equations with variable coefficients using weak Galerkin methods

A weak Galerkin method with rectangle partitions for the Signorini problem

A posteriori error estimates for weak Galerkin methods for second order elliptic problems on polygonal meshes

An over-penalized weak Galerkin method for parabolic interface problems with time-dependent coefficients

A newly developed weak Galerkin method is proposed to solve parabolic equations. This method allows the usage of totally discontinuous functions in approximation space and preserves the energy conservation law. Both continuous and discontinuous time weak Galerkin finite element schemes are developed and analyzed. Optimal‐order error estimates in both H1 and L2 norms are established. Numerical tests are performed and reported. © 2013 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2013