Family Of Discontinuous Galerkin Methods Research Articles

A new analysis of discontinuous Galerkin methods for a fourth order variational inequality

We study a family of discontinuous Galerkin methods for the displacement obstacle problem of Kirchhoff plates on two and three dimensional convex polyhedral domains, which are characterized as fourth order elliptic variational inequalities of the first kind. We prove that the error in an H2-like energy norm is O(hα) for the quadratic method, where α∈(12,1] is determined by the geometry of the domain. Under additional assumptions on the contact set such that the solution has improved regularity, we derive the optimal error estimate with α∈(1,32) for the cubic method. Numerical experiments demonstrate the performance of the methods and confirm the theoretical results.

A three-dimensional nodal-based implementation of a family of discontinuous Galerkin methods for elasticity problems

Four discontinuous Galerkin (DG) methods are proposed to enrich the resource of modeling elasticity problems as they are volume locking-free and allow hanging nodes in meshing. A detailed finite element formulation of these DG methods is presented. For implementation, we coded a three-dimensional nodal-based DG program in which the conventional nodal-based pure displacement finite element codes are fully exploited. The robustness and accuracy of each DG method are demonstrated and compared with mixed methods through solving a rubber beam problem. The coupled use of DG with continuous elements is proposed for some practical applications.

A coupling of mixed and discontinuous Galerkin finite-element methods for poroelasticity

In this paper, we formulate a finite-element procedure for approximating the coupled fluid and mechanics in Biot’s consolidation model of poroelasticity. We approximate the flow variables by a mixed finite-element space and the displacement by a family of discontinuous Galerkin methods. Theoretical convergence error estimates are derived and, in particular, are shown to be independent of the constrained specific storage coefficient, c o . This suggests that our proposed algorithm is a potentially effective way to combat locking, or the nonphysical pressure oscillations, which sometimes arise in numerical algorithms for poroelasticity.

Analysis of a family of discontinuous Galerkin methods for elliptic problems: the one dimensional case

In this paper we analyze a family of discontinuous Galerkin methods, parameterized by two real parameters, for elliptic problems in one dimension. Our main results are: (1) a complete inf-sup stability analysis characterizing the parameter values yielding a stable scheme and energy norm error estimates as a direct consequence thereof, (2) an analysis of the error in L2 where the standard duality argument only works for special parameter values yielding a symmetric bilinear form and different orders of convergence are obtained for odd and even order polynomials in the nonsymmetric case. The analysis is consistent with numerical results and similar behavior is observed in two dimensions.