This note describes an implementation of a discontinuous Petrov Galerkin (DPG) method for acoustic waves within the framework of high order finite elements provided by the software package NGSolve. A technique to impose the impedance boundary condition weakly is indicated. Numerical results from this implementation show that a multiplicative Schwarz algorithm, with no coarse solve, provides a p...

In this paper, we uncover and study a new superconvergence property of a large class of finite element methods for one-dimensional convectiondiffusion problems. This class includes discontinuous Galerkin methods defined in terms of numerical traces, discontinuous Petrov–Galerkin methods and hybridized mixed methods. We prove that the so-called numerical traces of both variables superconverge at...

In this paper, we combine continuous Galerkin-Petrov (cGP) and discontinuous Galerkin (dG) time stepping schemes with local projection method applied to inf-sup stable discretization of the transient Oseen problem. Using variational-type time-discretization methods of polynomial degree k, we show that the cGP(k) and dG(k) methods are accurate of order k+1, in the whole time interval. Moreover, ...

A new symmetric local projection method built on residual bases (RELP) makes linear equal-order finite element pairs stable for the Darcy problem. The derivation is performed inside a Petrov-Galerkin enriching space approach (PGEM) which indicates parameter-free terms to be added to the Galerkin method without compromising consistency. Velocity and pressure spaces are augmented using solutions ...

We reveal the relationship between a Petrov–Galerkin method and a spectral collocation method at the Chebyshev points of the second kind (±1 and zeros of Uk) for the two-point boundary value problem. Derivative superconvergence points are identified as the Chebyshev points of the first kind (Zeros of Tk). Super-geometric convergent rate is established for a special class of solutions.

We investigate the numerical solution of an integro-differential equation with a memory term. For the time discretization we apply the continuous Petrov–Galerkin method considered by Lin et al. [SIAM J. Numer. Anal., 38, 2000]. We combined the Petrov–Galerkin scheme with respect to time with continuous finite elements for the space discretization and obtained a fully discrete scheme. We show op...