Mixed Discontinuous Galerkin Finite Element Research Articles

Hybrid mixed discontinuous Galerkin finite element method for incompressible miscible displacement problem

A new hybrid mixed discontinuous Galerkin finite element (HMDGFE) method is constructed for incompressible miscible displacement problem. In this method, the hybrid mixed finite element (HMFE) procedure is considered to solve the pressure and velocity equations, and a new hybrid mixed discontinuous Galerkin finite element procedure is constructed to solve the concentration equation with upwind technique. Compared with other traditional discontinuous Galerkin methods, the new method can reach a global system with less unknowns and sparser stencils. The consistency and conservation of the method are analyzed, the stability and optimal error estimates are also derived by the new technique. Meanwhile, numerical examples are provided to confirm the corresponding theoretical analysis.

Hybrid mixed discontinuous Galerkin finite element method for incompressible wormhole propagation problem

Hybrid mixed discontinuous Galerkin finite element method for incompressible wormhole propagation problem

A hybrid mixed finite element method for convection-diffusion-reaction equation with local exponential fitting technique

A new hybrid mixed finite element method is proposed for solving the convection-diffusion-reaction equation with local exponential fitting technique. In this method, the exponential fitting technique in [7] is used to discretize the convection-diffusion-reaction equation by introducing a new variable at element level; then the hybrid mixed discontinuous Galerkin finite element method is used to approximate the discretized problem. The convergence of the proposed method is analyzed, and the corresponding a priori error estimate is derived. The numerical results are presented to confirm our theoretical analysis.

A posteriori error estimates of mixed discontinuous Galerkin method for a class of Stokes eigenvalue problems

<abstract><p>For a class of Stokes eigenvalue problems including the classical Stokes eigenvalue problem and the magnetohydrodynamic Stokes eigenvalue problem a residual type a posteriori error estimate of the mixed discontinuous Galerkin finite element method using $ \mathbb{P}_{k}-\mathbb{P}_{k-1} $ element $ (k\geq 1) $ is studied in this paper. The a posteriori error estimators for approximate eigenpairs are given. The reliability and efficiency of the posteriori error estimator for the eigenfunction are proved and the reliability of the estimator for the eigenvalue is also analyzed. The numerical results are provided to confirm the theoretical predictions and indicate that the method considered in this paper can reach the optimal convergence order $ O(dof^{\frac{-2k}{d}}) $.</p></abstract>

Stability and Convergence of Spectral Mixed Discontinuous Galerkin Methods for 3D Linear Elasticity on Anisotropic Geometric Meshes

We consider spectral mixed discontinuous Galerkin finite element discretizations of the Lame system of linear elasticity in polyhedral domains in $${\mathbb {R}}^3$$. In order to resolve possible corner, edge, and corner-edge singularities, anisotropic geometric edge meshes consisting of hexahedral elements are applied. We perform a computational study on the discrete inf-sup stability of these methods, and especially focus on the robustness with respect to the Poisson ratio close to the incompressible limit (i.e. the Stokes system). Furthermore, under certain realistic assumptions (for analytic data) on the regularity of the exact solution, we illustrate numerically that the proposed mixed DG schemes converge exponentially in a natural DG norm.

A priori and a posteriori error analysis for the mixed discontinuous Galerkin finite element approximations of the biharmonic problems

In this article, a new mixed discontinuous Galerkin finite element method is proposed for the biharmonic equation in two or three‐dimension space. It is amenable to an efficient implementation displaying new convergence properties. Through an auxiliary variable , we rewrite the problem into a two‐order system. Then, the a priori error estimates are derived in L2 norm and in the broken DG norm for both u and p. We prove that, when polynomials of degree r () are used, we obtain the optimal convergence rate of order r + 1 in L2 norm and of order r in DG norm for u, and the order r in both norms for . The numerical experiments illustrate the theoretic order of convergence. For the purpose of adaptive finite element method, the a posteriori error estimators are also proposed and proved to field a sharp upper bound. We also provide numerical evidence that the error estimators and indicators can effectively drive the adaptive strategies. © 2016 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 33: 318–353, 2017

Mixed Discontinuous Galerkin Time-Stepping Method for Linear Parabolic Optimal Control Problems

In this paper, we discuss the mixed discontinuous Galerkin (DG) finite element approximation to linear parabolic optimal control problems. For the state variables and the co-state variables, the discontinuous finite element method is used for the time discretization and the Raviart-Thomas mixed finite element method is used for the space discretization. We do not discretize the space of admissible control but implicitly utilize the relation between co-state and control for the discretization of the control. We derive a priori error estimates for the lowest order mixed DG finite element approximation. Moveover, for the element of arbitrary order in space and time, we derive a posteriori L 2 (0,T;L 2 ()) error estimates for the scalar functions, assuming that only the underlying mesh is static. Finally, we present an example to confirm the theoretical result on a priori error estimates.

A mixed discontinuous Galerkin, convex splitting scheme for a modified Cahn-Hilliard equation and an efficient nonlinear multigrid solver

In this paper we devise and analyze a mixed discontinuous Galerkin finite element method for a modified Cahn-Hilliard equation that models phase separation in diblock copolymer melts. The time discretization is based on a convex splitting of the energy of the equation. We prove that our scheme is unconditionally energy stable with respect to a spatially discrete analogue of the continuous free energy of the system, unconditionally uniquely solvable, and convergent in the natural energy norm with optimal rates. We describe an efficient nonlinear multigrid solver for advancing our semi-implicit scheme in time and conclude the paper with numerical tests confirming the predicted convergence rates and suggesting the near-optimal complexity of the solver.

Energy Norm shape A Posteriori Error Estimation for Mixed Discontinuous Galerkin Approximations of the Stokes Problem

In this paper, we develop the a posteriori error estimation of mixed discontinuous Galerkin finite element approximations of the Stokes problem. In particular, we derive computable upper bounds on the error, measured in terms of a natural (mesh-dependent) energy norm. This is done by rewriting the underlying method in a non-consistent form using appropriate lifting operators, and by employing a decomposition result for the discontinuous spaces. A series of numerical experiments highlighting the performance of the proposed a posteriori error estimator on adaptively refined meshes are presented.

Energy norm a posteriori error estimation for mixed discontinuous Galerkin approximations of the stokes problem

In this paper, we develop the a posteriori error estimation of mixed discontinuous Galerkin finite element approximations of the Stokes problem. In particular, we derive computable upper bounds on ...

Energy norm a posteriori error estimation for mixed discontinuous Galerkin approximations of the Maxwell operator

In this paper we develop the a posteriori error estimation of mixed discontinuous Galerkin finite element approximations of the Maxwell operator. In particular, by employing suitable Helmholtz decompositions of the error, together with the conservation properties of the underlying method, computable upper bounds on the error, measured in terms of a natural (mesh-dependent) energy norm, are derived. Numerical experiments testing the performance of our a posteriori error bounds for problems with both smooth and singular analytical solutions are presented.