It has been observed from the authors’ numerical experiments (2007) that the Local Discontinuous Galerkin (LDG) method converges uniformly under the Shishkin mesh for singularly perturbed two-point boundary problems of the convection-diffusion type. Especially when using a piecewise polynomial space of degree k, the LDG solution achieves the optimal convergence rate k+1 under the L2-norm, and a...

This paper is concerned with automatic enrichment in the particlepartition of unity method (PPUM). The goal of our automatic enrichment is to recover the optimal convergence rate of the uniform h-version independent of the regularity of the solution. Hence, we employ enrichment not only for modeling purposes but rather to improve the approximation properties of the numerical scheme. To this end...

Here we propose a new method based on projections for the approximate solution of eigenvalue problems. For an integral operator with a smooth kernel, using an interpolatory projection at Gauss points onto the space of (discontinuous) piecewise polynomials of degree ≤ r−1, we show that the proposed method exhibits an error of the order of 4r for eigenvalue approximation and of the order of 3r fo...

An improved version of the multimodal admittance method in acoustic waveguides with varying cross sections is presented. This method aims at a better convergence with respect to the number of transverse modes that are taken into account. It is based on an enriched modal expansion of the pressure: the N first modes are the local transverse modes and a supplementary (N+1)th mode, called boundary ...

Discontinuous Galerkin (DG) methods exhibit ”hidden accuracy” that makes the superconvergence of this method an increasing popular topic to address. Previous work has implemented a convolution kernel approach that allows us to improve the order of accuracy from k+1 to order 2k+m for time-dependent linear convection-diffusion equations, where k is the highest degree polynomial used in the approx...

We propose a simple quantum algorithm for simulating highly oscillatory dynamics, which does not require complicated control logic handling time-ordering operators. To our knowledge, this is the first that both insensitive to rapid changes of time-dependent Hamiltonian and exhibits commutator scaling. Our method can be used efficient simulation in interaction picture. In particular, we demonstr...

In this paper, we study the superconvergence behavior of discontinuous Galerkin methods using upwind numerical fluxes for one-dimensional linear hyperbolic equations with degenerate variable coefficients. The study establishes superconvergence results for the flux function approximation as well as for the DG solution itself. To be more precise, we first prove that the DG flux function is superc...

We consider mixed nite element methods for second order elliptic equations on non-matching multiblock grids. A mortar nite element space is introduced on the non-matching interfaces. We approximate in this mortar space the trace of the solution, and we impose weakly a continuity of ux condition. A standard mixed nite element method is used within the blocks. Optimal order convergence is shown f...

We develop a family of inexpensive discretization schemes for diffusion problems on generalized polyhedral meshes with elements having non-planar faces. The material properties are described by a full tensor. We also prove superconvergence for the scalar (pressure) variable under very general assumptions. The theoretical results are confirmed with numerical experiments. In the practically impor...

In the present paper, we discuss the superconvergence of the interpolated Galerkin solutions for Hammerstein equations. With the interpolation post-processing for the Galerkin approximation xh, we get a higher order approximation I 2r−1 2h xh, whose convergence order is the same as that of the iterated Galerkin solution. Such an interpolation post-processing method is much simpler than the iter...