We prove error estimates for the ultra weak variational formulation (UWVF) in 3D linearelasticity. We show that the UWVF of Navier’s equation can be derived as an upwinddiscontinuous Galerkin method. Using this observation, error estimates are investigatedapplying techniques from the theory of discontinuous Galerkin methods. In particular, wederive a basic error estimate for the UWVF in a discontinuous Galerkin type norm and thenan error estimate in the L2(Ω) norm in terms of the bestapproximation error. Our final result is an L2(Ω) norm errorestimate using approximation properties of plane waves to give an estimate for the orderof convergence. Numerical examples are presented.

We investigate the ultra weak variational formulation (UWVF) of the 2-D Helmholtz equation using a new choice of basis functions. Traditionally the UWVF basis functions are chosen to be plane waves. Here, we instead use first kind Bessel functions. We compare the performance of the two bases. Moreover, we show that it is possible to use coupled plane wave and Bessel bases in the same mesh. As test cases we shall consider propagating plane and evanescent waves in a rectangular domain and a singular 2-D Helmholtz problem in an L-shaped domain.