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Register a new userAn Analysis of the Weak Finite Element Method for Convection-Diffusion Equations
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We study the weak finite element method solving convection-diffusion equations. A weak finite element scheme is presented based on a spacial variational form. We established a weak embedding inequality that is very useful in the weak finite element analysis. The optimal order error estimates are derived in the discrete $H^1$-norm, the $L_2$-norm and the $L_infty$-norm, respectively. In particular, the $H^1$-superconvergence of order $k+2$ is given under certain condition. Finally, numerical examples are provided to illustrate our theoretical analysis
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A two-step preconditioned iterative method based on the Hermitian/Skew-Hermitian splitting is applied to the solution of nonsymmetric linear systems arising from the Finite Element approximation of convection-diffusion equations. The theoretical spectral analysis focuses on the case of matrix sequences related to FE approximations on uniform structured meshes, by referring to spectral tools derived from Toeplitz theory. In such a setting, if the problem is coercive, and the diffusive and convective coefficients are regular enough, then the proposed preconditioned matrix sequence shows a strong clustering at unity, i.e., a superlinear preconditioning sequence is obtained. Under the same assumptions, the optimality of the PHSS method is proved and some numerical experiments confirm the theoretical results. Tests on unstructured meshes are also presented, showing the some convergence behavior.
The paper is devoted to the spectral analysis of effective preconditioners for linear systems obtained via a Finite Element approximation to diffusion-dominated convection-diffusion equations. We consider a model setting in which the structured finite element partition is made by equi-lateral triangles. Under such assumptions, if the problem is coercive, and the diffusive and convective coefficients are regular enough, then the proposed preconditioned matrix sequences exhibit a strong clustering at unity, the preconditioning matrix sequence and the original matrix sequence are spectrally equivalent, and the eigenvector matrices have a mild conditioning. The obtained results allow to show the optimality of the related preconditioned Krylov methods. %It is important to stress that The interest of such a study relies on the observation that automatic grid generators tend to construct equi-lateral triangles when the mesh is fine enough. Numerical tests, both on the model setting and in the non-structured case, show the effectiveness of the proposal and the correctness of the theoretical findings.
We propose an energy-stable parametric finite element method (ES-PFEM) to discretize the motion of a closed curve under surface diffusion with an anisotropic surface energy $gamma(theta)$ -- anisotropic surface diffusion -- in two dimensions, while $theta$ is the angle between the outward unit normal vector and the vertical axis. By introducing a positive definite surface energy (density) matrix $G(theta)$, we present a new and simple variational formulation for the anisotropic surface diffusion and prove that it satisfies area/mass conservation and energy dissipation. The variational problem is discretized in space by the parametric finite element method and area/mass conservation and energy dissipation are established for the semi-discretization. Then the problem is further discretized in time by a (semi-implicit) backward Euler method so that only a linear system is to be solved at each time step for the full-discretization and thus it is efficient. We establish well-posedness of the full-discretization and identify some simple conditions on $gamma(theta)$ such that the full-discretization keeps energy dissipation and thus it is unconditionally energy-stable. Finally the ES-PFEM is applied to simulate solid-state dewetting of thin films with anisotropic surface energies, i.e. the motion of an open curve under anisotropic surface diffusion with proper boundary conditions at the two triple points moving along the horizontal substrate. Numerical results are reported to demonstrate the efficiency and accuracy as well as energy dissipation of the proposed ES-PFEM.
In this paper, we present an embedded staggered discontinuous Galerkin method for the convection-diffusion equation. The new method combines the advantages of staggered discontinuous Galerkin (SDG) and embedded discontinuous Galerkin (EDG) method, and results in many good properties, namely local and global conservations, free of carefully designed stabilization terms or flux conditions and high computational efficiency. In applying the new method to convection-dominated problems, the method provides optimal convergence in potential and suboptimal convergence in flux, which is comparable to other existing DG methods, and achieves $L^2$ stability by making use of a skew-symmetric discretization of the convection term, irrespective of diffusivity. We will present numerical results to show the performance of the method.
We propose a weak Galerkin(WG) finite element method for solving the one-dimensional Burgers equation. Based on a new weak variational form, both semi-discrete and fully-discrete WG finite element schemes are established and analyzed. We prove the existence of the discrete solution and derive the optimal order error estimates in the discrete $H^1$-norm and $L^2$-norm, respectively. Numerical experiments are presented to illustrate our theoretical analysis.
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