We study the convergence behavior of collocation schemes applied to approximate solutions of BVPs in linear index 1 DAEs which exhibit a critical point at the left boundary. Such a critical point of the DAE causes a singularity within the inherent ODE system. We focus our attention on the case when the inherent ODE system is singular with a singularity of the first kind, apply polynomial colloc...

We study numerical methods for sampling probability measures in high dimension where the underlying model is only approximately identified with a gradient system. Extended stochastic dynamical methods are discussed which have application to multiscale models, nonequilibrium molecular dynamics, and Bayesian sampling techniques arising in emerging machine learning applications. In addition to pro...

This paper proposes and analyzes a stabilized finite-volume method FVM for the threedimensional stationary Navier-Stokes equations approximated by the lowest order finite element pairs. The method studies the new stabilized FVM with the relationship between the stabilized FEM FEM and the stabilized FVM under the assumption of the uniqueness condition. The results have three prominent features i...

In this paper, we study the convergence and time evolution of the error between the discontinuous Galerkin (DG) finite element solution and the exact solution for conservation laws when upwind fluxes are used. We prove that if we apply piecewise linear polynomials to a linear scalar equation, the DG solution will be superconvergent towards a particular projection of the exact solution. Thus, th...

Abstract A frequency accuracy study is presented for the isogeometric free vibration analysis of Mindlin–Reissner plates using reduced integration and quadratic splines, which reveals an interesting coarse mesh superconvergence. Firstly, error estimates discretization with splines are rationally derived, where degeneration to Timoshenko beams discussed as well. Subsequently, in accordance these...

The purpose of this paper is twofold: to study the superconvergence properties and present an efficient reliable a posteriori error estimator for local discontinuous Galerkin (LDG) method linear second-order elliptic problems on Cartesian grids. We prove that LDG solution superconvergent towards particular projection exact solution. order convergence proved be p + 2 , when tensor product polyno...

Abstract. In this paper, we provide the optimal convergence rate of a posteriori error estimates for the local discontinuous Galerkin (LDG) method for the second-order wave equation in one space dimension. One of the key ingredients in our analysis is the recent optimal superconvergence result in [W. Cao, D. Li and Z. Zhang, Commun. Comput. Phys. 21 (1) (2017) 211-236]. We first prove that the ...

Local projection methods which yield c'm_1) piecewise polynomials of order m + k as approximate solutions of a boundary value problem for an mth order ordinary differential equation are determined by the k linear functional at which the residual error in each partition interval is required to vanish on. We develop a condition on these k f unctionals which implies breakpoint superconvergence (of...

We propose an arbitrary-order accurate Mimetic Finite Difference (MFD) method for the approximation of diffusion problems in mixed form on unstructured polygonal and polyhedral meshes. As usual in the mimetic numerical technology, the method satisfies local consistency and stability conditions, which determines the accuracy and the well-posedness of the resulting approximation. The method also ...

In this paper, we derive improved a priori error estimates for families of hybridizable interior penalty discontinuous Galerkin (H-IP) methods using variable second-order elliptic problems. The strategy is to use penalization function the form O ( 1 / h + δ ) , where denotes mesh size and user-dependent parameter. We then quantify its direct impact on convergence analysis, namely, (strong) cons...