A Priori Error Estimates for Finite Element Methods Based on Discontinuous Approximation Spaces for Elliptic Problems

Abstract

We analyze three discontinuous Galerkin approximations for solving elliptic problems in two or three dimensions. In each one, the basic bilinear form is nonsymmetric: the first one has a penalty term on edges, the second has one constraint per edge, and the third is totally unconstrained. For each of them we prove hp error estimates in the H 1 norm, optimal with respect to h, the mesh size, and nearly optimal with respect to p, the degree of polynomial approximation. We establish these results for general elements in two and three dimensions. For the unconstrained method, we establish a new approximation result valid on simplicial elements. L 2 estimates are also derived for the three methods. Introduction.Over the past five years, discontinuous Galerkin methods have become very widely used for solving a large range of computational fluid problems. They are preferred over more standard continuous methods because of their flexibility in approximating globally rough solutions, their local mass conservation, their possible definition on unstructured meshes, their potential for error control and mesh adaptation, and their straightforward applications to anisotropic materials. Another interesting aspect of these methods is that they can employ polynomials of degree k (P k ) on quadrilaterals, whose dimension is substantially lower than that of Q k , the standard space of functions on quadrilaterals. Similarly, they lead to smaller systems of equations than the locally conservative mixed finite element methods and do not require the solution of saddle-point problems.In this paper, we discuss three numerical algorithms for elliptic problems which employ discontinuous approximation spaces. The three methods are called the nonsymmetric interior penalty Galerkin (NIPG) method, the nonsymmetric constrained Galerkin (NCG) method, and the discontinuous Galerkin (DG) method. The three algorithms are closely related in that the underlying bilinear form for all three is the same and is nonsymmetric.In the NIPG method, we modify the bilinear form of the interior penalty Galerkin method treated by Douglas and Dupont [9], Wheeler [18], Arnold [2], and Wheeler and Darlow [19]. Here, we antisymmetrize the bilinear form by changing the sign of one term and as a consequence we require only a positive penalty, whereas the proofs in [18] assume that the penalty is bounded below by a problem-dependent constant. We choose the penalty term in such a way that we can establish both H 1 and L 2 optimal convergence rates.

A parameter-free variational coupling approach for trimmed isogeometric thin shells

“…Therefore, it does not require the introduction of additional stabilization terms with associated parameters and, in contrast to the symmetric form of Nitsche's method, its performance does not depend on the accuracy of the variational estimate or the reliability and robustness of associated numerical algorithms. On the other hand, the non-symmetric Nitsche method leads to unsymmetric system matrices and its numerical analysis framework does not cover optimal convergence rates of the L 2 error [54][55][56][57][58]. This paper extends recent work [59][60][61] that demonstrated the potential of the non-symmetric Nitsche method for parameter-free analysis in the context of non-matching and non-boundary-fitted discretizations.…”

“…In the context of the non-symmetric interior penalty discontinous Galerkin method [55,57], penalty-free nonsymmetric Nitsche formulations have been reported to lead to oscillations near interfaces [77]. One way to effectively reduce these oscillations is to introduce a stabilization term, that has the same form as in the symmetric Nitsche method.…”

“…In particular, the average operator for vector quantities is defined in (18), t denotes the shell thickness, and the terms in brackets correspond to definitions (53) to (57). We note that the stabilization terms of equation (76) refer to the global Cartesian basis.…”

Heating of Semiconductor Devices in Electric Circuits

“…More specifically, we present an example for which the suboptimal rate of convergence with respect to the polynomial degree is both proven theoretically and validated through numerical experiments; hence, the known a-priori bounds from the literature [11,9] are sharp, i.e., the p-IPDG method is indeed suboptimal by half an order of p.…”

“…To the best of our knowledge, the sharpest known general error bounds (in the Hilbertian Sobolev space setting) for the hp-version interior penalty DG method for second-order elliptic PDEs are due to Rivière, Wheeler and Girault [11] and Houston, Schwab and Süli [9]; when the error is measured in the (natural) energy norm, the a-priori bounds are optimal with respect to the meshsize h but are suboptimal with respect to the polynomial degree p by half an order of p.…”