Error estimates for the ultra weak variational formulation in linear elasticity

Abstract

Abstract. We prove error estimates for the ultra weak variational formulation (UWVF) in 3D linear elasticity. We show that the UWVF of Navier's equation can be derived as an upwind discontinuous Galerkin method. Using this observation, error estimates are investigated applying techniques from the theory of discontinuous Galerkin methods. In particular, we derive a basic error estimate for the UWVF in a discontinuous Galerkin type norm and then an error estimate in the L 2 (Ω) norm in terms of the best approximation error. Our final result is an L 2 (Ω) norm error estimate using approximation properties of plane waves to give an estimate for the order of convergence. Numerical examples are presented.Mathematics Subject Classification. 65N15, 65N30, 74J05, 74S30.

The equal spacing of N points on a sphere with application to partition-of-unity wave diffraction problems

“…The choice of such a set of directions, being a set of unit vectors on the unit sphere, presents a directly equivalent problem. Confining this discussion to the various three-dimensional wave diffraction algorithms that exist, an efficient spacing algorithm would be of benefit in the plane-wave methods of Perrey-Debain et al [2], in the discontinuous enrichment method of Massimi et al [3], in the variational theory of complex rays of Kovalevsky et al [4], in the ultra weak variational formulation of Luostari et al [5], as well as other Trefftz methods. It is not the intention of the authors to present in this paper a detailed review of such methods; the interested reader is referred to Bettess [6].…”

A Trefftz-discontinuous Galerkin method for time-harmonic elastic wave problems

“…To prove an error estimate in L 2 ( ) for the PWDG method, we adopt the approach from Buffa and Monk (2008), Cummings and Feng (2006), Hiptmair et al (2011), Luostari et al (2013). Considering the dual (nonhomogeneous) problem of the Navier equation (2.1) (see Cummings and Feng 2006)…”

A plane wave discontinuous Galerkin method with a Dirichlet-to-Neumann boundary condition for the scattering problem in acoustics

“…These methods include the Partition of Unity Method (PUM) of Melenk and Babuska, [2] the Discontinuous Enrichment Method (DEM) [11], the Ultra Weak Variational Formulation (UWVF) of Cessenat and Després [6]. The UWVF has been applied to the Maxwell equations [20], linear elasticity [25], acoustic fluid-solid interaction [19] and to thin clamped plate problems [26]. More recently, the Plane Wave Discontinuous Galerkin (PWDG) method has been studied by Hiptmair et al [15,16,17,12] as a generalization of the UWVF method.…”