We propose an arbitrary-order discontinuous Galerkin method for secondorder elliptic problem on general polygonal mesh with only one degree of freedom per element. This is achieved by locally solving a discrete least-squares over a neighboring element patch. Under a geometrical condition on the element patch, we prove an optimal a priori error estimates for the energy norm and for the L 2 norm. The accuracy and the efficiency of the method up to order six on several polygonal meshes are illustrated by a set of benchmark problems.
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. keyword: Least-squares problem · Reconstructed basis function · Discontinuous Galerkin method · Biharmonic problem MSC2010: 49N45; 65N21
In this paper, we develop a patch reconstruction finite element method for the Stokes problem. The weak formulation of the interior penalty discontinuous Galerkin is employed. The proposed method has a great flexibility in velocitypressure space pairs whose stability properties are confirmed by the inf-sup tests. Numerical examples show the applicability and efficiency of the proposed method.Galerkin method · Inf-sup test 2010 MSC: 49N45, 65N21 IntroductionWe are concerned in this paper with the incompressible Stokes problem, which has a wide range of applications on the approximation of low Reynolds number flows and the time discretizations of the Oseen equation or Naiver-Stokes equation. One of the major difficulties in finite element discretizations for the Stokes problem is the incompressible constraint, which leads to a saddlepoint problem. The stability condition often referred as the inf-sup (LBB) condition requires the approximation spaces for velocity and pressure need to be carefully chosen [1]. We refer to [2,3] for some specific spaces used in the traditional finite element methods to solve the Stokes problem.Most recently, the discontinuous Galerkin (DG) methods have achieved a great success in computational fluid dynamics, see the state of art survey [4].Hansbo and Larson propose and analyze an interior penalty DG method for incompressible and nearly incompressible linear elasticity on triangular meshes in [5] where polynomial spaces of degree k and k − 1 are employed to approximate velocity and pressure, respectively. In [6] Toselli considers the hpapproximations for the Stokes problem using piecewise polynomial spaces. The uniform divergence stability and error estimates with respect to h and p are proven for this DG formulation when velocity is approximated one or two degrees higher than pressure. Numerical results show that using equal order spaces for velocity and pressure can also work well. Schötzau et al. improve the estimates on tensor product meshes in [7]. A local discontinuous Galerkin method (LDG) for the Stokes problem is proposed in [8]. The LDG method can be considered as a stabilized method when the approximation spaces for velocity and pressure are chosen with the same order. Hybrid discontinuous Galerkin methods are also of interest due to their capability of providing a superconvergent post processing, we refer to [9,10,11,12] for more discussion.Some special finite element spaces can be adopted to Stokes problem in DG framework. Karakashian and his coworkers [13,14] propose a DG method with piecewise solenoidal vector fields which are locally divergence-free. Cockburn et al. [15,16,9,17] develop the LDG method with solenoidal vector fields. By introducing the hybrid pressure, the pressure and the globally divergence-free velocity can be obtained by a post-process of the LDG solution. While Montlaur et al. [18] present two DG formulations for the incompressible flow, the first formulation is derived from an interior penalty method such that the computation of the velocity and t...
We adapt a symmetric interior penalty discontinuous Galerkin method using a patch reconstructed approximation space to solve elliptic eigenvalue problems, including both second and fourth order problems in 2D and 3D. It is a direct extension of the method recently proposed to solve corresponding boundary value problems, and the optimal error estimates of the approximation to eigenfunctions and eigenvalues are instant consequences from existing results. The method enjoys the advantage that it uses only one degree of freedom on each element to achieve very high order accuracy, which is highly preferred for eigenvalue problems as implied by Zhang's recent study [J. Sci. Comput. 65 (2), 2015]. By numerical results, we illustrate that higher order methods can provide much more reliable eigenvalues. To justify that our method is the right one for eigenvalue problems, we show that the patch reconstructed approximation space attains the same accuracy with fewer degrees of freedom than classical discontinuous Galerkin methods. With the increasing of the polynomial order, our method can even achieve a better performance than conforming finite element methods, such methods are traditionally the methods of choice to solve problems with high regularities. keyword: elliptic eigenvalue problem, discontinuous Galerkin method, patch reconstruction MSC2010: 49N45; 65N21 arXiv:1901.01803v2 [math.NA]
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