Sharp Expressions for the Stabilization Parameters in Symmetric Interior-penalty Discontinuous Galerkin Finite Element Approximations of Fourth-order Elliptic Problems

Abstract

-In this paper, we derive explicit expressions for the penalty parameters appearing in symmetric and semi-symmetric interior-penalty discontinuous Galerkin finite element method (DGFEM) for fourth-order elliptic problems. We demonstrate the sharpness of the theoretically predicted penalty parameter values through numerical experiments.2000 Mathematics Subject Classification: 65N12, 65N30.Keywords: discontinuous Galerkin finite element method, stabilized finite element methods, error bounds, stability and convergence.

Discontinuous Galerkin Finite Element Approximation of the Cahn–Hilliard Equation with Convection

“…An explicit expression for the penalty parameter α 0 in the interior-penalty discontinuous Galerkin finite element approximation of a second-order elliptic problem was proposed by Shabhazi [55] for meshes consisting of simplicial elements. The explicit dependence of the coercivity constant C 0 on the polynomial degree and the angles of the triangular/quadrilateral mesh elements was derived by Epshteyn and Rivière [33]; Mozolevski and Bösing [46] derived explicit expressions for penalty parameters in symmetric interior-penalty discontinuous Galerkin approximations of fourth-order elliptic problems on meshes consisting of parallelepipeds. 4.…”

Optimal penalty parameter forC0IPDG

“…In particular, they showed that the parameter depends on the smallest cot θ over all angles of the triangle in 2D or over all dihedral angles in the tetrahedron in 3D. For further study on the penalty problem, we refer the readers to [5][6][7][8][9][10][11].…”

A Superconvergent $$C^0$$ C 0 Discontinuous Galerkin Method for Kirchhoff Plates: Error Estimates, Hybridization and Postprocessing

“…It is noted that some dimensionless penalty parameters presented in symmetric fully discontinuous IP methods and C 0 IP methods must be chosen large enough to guarantee stability, which can not be precisely quantified a priori, yielding some difficulty in practical applications. To resolve this problem, several explicit expressions for penalty parameters in symmetric IP methods on parallelepiped partitions were derived in [38]. Through introducing lifting operators, Wells and Dung proposed in [45] a C 0 DG (CDG) method whose penalty coefficients can be precisely quantified a priori.…”