A conforming discontinuous Galerkin finite element method on rectangular partitions

Feng, Yue; Liu, Yujie; Wang, Ruishu; Zhang, Shangyou

doi:10.3934/era.2020120

Cited by 10 publications

(6 citation statements)

References 19 publications

(38 reference statements)

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“…This allows for recent developments such that, e.g. , conforming discontinuous Galerkin elements on rectangular mesh [23], or even for meshfree methods [45]. Next, Section 5 shall focus on well-known examples of suitable choices of such families of conforming finite elements.…”

Section: Numerical Analysismentioning

confidence: 99%

Numerical Analysis of a Structure-Preserving Space-Discretization for an Anisotropic and Heterogeneous Boundary Controlled $N$-Dimensional Wave Equation As a Port-Hamiltonian System

Haine¹,

Matignon²,

Serhani³

2023

IJNAM

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The anisotropic and heterogeneous N -dimensional wave equation, controlled and observed at the boundary, is considered as a port-Hamiltonian system. A recent structure-preserving mixed Galerkin method is applied, leading directly to a finite-dimensional port-Hamiltonian system: its numerical analysis is carried out in a general framework. Optimal choices of mixed finite elements are then proved to reach the best trade-off between the convergence rate and the number of degrees of freedom for the state error. Exta compatibility conditions are identified for the Hamiltonian error to be twice that of the state error, and numerical evidence is provided that some combinations of finite element families meet these conditions. Numerical simulations are performed in 2D to illustrate the main theorems among several choices of classical finite element families. Several test cases are provided, including non-convex domain, anisotropic or heterogeneous cases and absorbing boundary conditions.

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Section: Numerical Analysismentioning

confidence: 99%

Numerical Analysis of a Structure-Preserving Space-Discretization for an Anisotropic and Heterogeneous Boundary Controlled $N$-Dimensional Wave Equation As a Port-Hamiltonian System

Haine¹,

Matignon²,

Serhani³

2023

IJNAM

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“…Anal. 19 (4), 742-760 (1982) [3] Cui, M., Zhang, S.: On the uniform convergence of the weak Galerkin finite element method for a singularly-perturbed biharmonic equation. J. Sci.…”

Section: R E F E R E N C E Smentioning

confidence: 99%

A superconvergent CDG finite element for the Poisson equation on polytopal meshes

Ye,

Zhang

2023

Z Angew Math Mech

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A conforming discontinuous Galerkin (CDG) finite element is constructed for solving second order elliptic equations on polygonal and polyhedral meshes. The numerical trace on the edge between two elements is no longer the average of two discontinuous Pk functions on the two sides, but a lifted function from four Pk functions. When the numerical gradient space is the subspace of piecewise polynomials on subtriangles/subtehrahedra of a polygon/polyhedron T which have a one‐piece polynomial divergence on T, this CDG method has a superconvergence of order two above the optimal order. Due to the superconvergence, we define a post‐process which lifts a Pk CDG solution to a quasi‐optimal solution on each element. Numerical examples in 2D and 3D are computed and the results confirm the theory.

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“…The CDG method is derived from the WG method by eliminating the degree of freedoms defined on element boundary. The CDG finite element method, introduced in [28], has been studied for diffusion problem [7,29], and for biharmonic equations [30].…”

Section: Introductionmentioning

confidence: 99%

Order Two Superconvergence of the CDG Finite Elements on Triangular and Tetrahedral Meshes

Ye¹,

Zhang²

2023

CSIAM-AM

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It is known that discontinuous finite element methods use more unknown variables but have the same convergence rate comparing to their continuous counterpart. In this paper, a novel conforming discontinuous Galerkin (CDG) finite element method is introduced for Poisson equation using discontinuous P k elements on triangular and tetrahedral meshes. Our new CDG method maximizes the potential of discontinuous P k element in order to improve the convergence rate. Superconvergence of order two for the CDG finite element solution is proved in an energy norm and in the L 2 norm. A local post-process is defined which lifts a P k CDG solution to a discontinuous P k+2 solution. It is proved that the lifted P k+2 solution converges at the optimal order. The numerical tests confirm the theoretic findings. Numerical comparison is provided in 2D and 3D, showing the P k CDG finite element is as good as the P k+2 continuous Galerkin finite element.

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A conforming discontinuous Galerkin finite element method on rectangular partitions

Cited by 10 publications

References 19 publications

Numerical Analysis of a Structure-Preserving Space-Discretization for an Anisotropic and Heterogeneous Boundary Controlled $N$-Dimensional Wave Equation As a Port-Hamiltonian System

Numerical Analysis of a Structure-Preserving Space-Discretization for an Anisotropic and Heterogeneous Boundary Controlled $N$-Dimensional Wave Equation As a Port-Hamiltonian System

A superconvergent CDG finite element for the Poisson equation on polytopal meshes

Order Two Superconvergence of the CDG Finite Elements on Triangular and Tetrahedral Meshes

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