Superconvergence analysis of spectral volume methods for one-dimensional diffusion and third-order wave equations

Xu Yin; Waixiang Cao; Zhimin Zhang

Displaying similar documents to “Superconvergence analysis of spectral volume methods for one-dimensional diffusion and third-order wave equations”

Theoretical and numerical studies of the $P_{N} P_{M}$ DG schemes in one space dimension

Abdulatif Badenjki, Gerald G. Warnecke (2019)

Applications of Mathematics

Similarity:

We give a proof of the existence of a solution of reconstruction operators used in the $P_{N} P_{M}$ DG schemes in one space dimension. Some properties and error estimates of the projection and reconstruction operators are presented. Then, by applying the $P_{N} P_{M}$ DG schemes to the linear advection equation, we study their stability obtaining maximal limits of the Courant numbers for several $P_{N} P_{M}$ DG schemes mostly experimentally. A numerical study explains how the stencils used in the reconstruction affect...

Unified error analysis of discontinuous Galerkin methods for parabolic obstacle problem

Papri Majumder (2021)

Applications of Mathematics

Similarity:

We introduce and study various discontinuous Galerkin (DG) finite element approximations for a parabolic variational inequality associated with a general obstacle problem in $ℝ^{d}$ $(d = 2, 3)$ . For the fully-discrete DG scheme, we employ a piecewise linear finite element space for spatial discretization, whereas the time discretization is carried out with the implicit backward Euler method. We present a unified error analysis for all well known symmetric and non-symmetric DG fully discrete...

An adaptive $h p$ -discontinuous Galerkin approach for nonlinear convection-diffusion problems

Dolejší, Vít

Similarity:

We deal with a numerical solution of nonlinear convection-diffusion equations with the aid of the discontinuous Galerkin method (DGM). We propose a new $h p$ -adaptation technique, which is based on a combination of a residuum estimator and a regularity indicator. The residuum estimator as well as the regularity indicator are easily evaluated quantities without the necessity to solve any local problem and/or any reconstruction of the approximate solution. The performance of the proposed $h p$ -DGM...

Explicit estimates for the summatory function of Λ(n)/n from the one of Λ(n)

Olivier Ramaré (2013)

Acta Arithmetica

Similarity:

We prove that the error term $\sum_{n \leq x} Λ (n) / n - l o g x + γ$ differs from (ψ(x)-x)/x by a well controlled function. We deduce very precise numerical results from the formula obtained.

Optimal observability of the multi-dimensional wave and Schrödinger equations in quantum ergodic domains

Yannick Privat, Emmanuel Trélat, Enrique Zuazua (2016)

Journal of the European Mathematical Society

Similarity:

We consider the wave and Schrödinger equations on a bounded open connected subset $Ω$ of a Riemannian manifold, with Dirichlet, Neumann or Robin boundary conditions whenever its boundary is nonempty. We observe the restriction of the solutions to a measurable subset $ω$ of $Ω$ during a time interval $[0, T]$ with $T > 0$ . It is well known that, if the pair $(ω, T)$ satisfies the Geometric Control Condition ( $ω$ being an open set), then an observability inequality holds guaranteeing that the total energy of solutions...

On the spectral radius in $L_{1} (G)$

E. Porada (1971)

Colloquium Mathematicae

Similarity:

$h p$ -anisotropic mesh adaptation technique based on interpolation error estimates

Dolejší, Vít

Similarity:

We present a completely new $h p$ -anisotropic mesh adaptation technique for the numerical solution of partial differential equations with the aid of a discontinuous piecewise polynomial approximation. This approach generates general anisotropic triangular grids and the corresponding degrees of polynomial approximation based on the minimization of the interpolation error. We develop the theoretical background of this approach and present a numerical example demonstrating the efficiency of...

Spectral projections for the twisted Laplacian

Herbert Koch, Fulvio Ricci (2007)

Studia Mathematica

Similarity:

Let n ≥ 1, d = 2n, and let (x,y) ∈ ℝⁿ × ℝⁿ be a generic point in ℝ²ⁿ. The twisted Laplacian $L = - 1 / 2 \sum_{j = 1}^{n} [(\partial_{x_{j}} + i y_{j}) ² + (\partial_{y_{j}} - i x_{j}) ²]$ has the spectrum n + 2k = λ²: k a nonnegative integer. Let $P_{λ}$ be the spectral projection onto the (infinite-dimensional) eigenspace. We find the optimal exponent ϱ(p) in the estimate $| | P_{λ} {u | |}_{L^{p} (ℝ^{d})} ≲ λ^{ϱ (p)} {| | u | |}_{L ² (ℝ^{d})}$ for all p ∈ [2,∞], improving previous partial results by Ratnakumar, Rawat and Thangavelu, and by Stempak and Zienkiewicz. The expression for ϱ(p) is ϱ(p) = 1/p -1/2 if 2 ≤ p ≤ 2(d+1)/(d-1), ϱ(p) = (d-2)/2 - d/p...

Face-to-face partition of 3D space with identical well-centered tetrahedra

Radim Hošek (2015)

Applications of Mathematics

Similarity:

The motivation for this paper comes from physical problems defined on bounded smooth domains $Ω$ in 3D. Numerical schemes for these problems are usually defined on some polyhedral domains $Ω_{h}$ and if there is some additional compactness result available, then the method may converge even if $Ω_{h} \to Ω$ only in the sense of compacts. Hence, we use the idea of meshing the whole space and defining the approximative domains as a subset of this partition. Numerical schemes for which quantities are defined...

Spectral synthesis and operator synthesis

K. Parthasarathy, R. Prakash (2006)

Studia Mathematica

Similarity:

Relations between spectral synthesis in the Fourier algebra A(G) of a compact group G and the concept of operator synthesis due to Arveson have been studied in the literature. For an A(G)-submodule X of VN(G), X-synthesis in A(G) has been introduced by E. Kaniuth and A. Lau and studied recently by the present authors. To any such X we associate a $V^{\infty} (G)$ -submodule X̂ of ℬ(L²(G)) (where $V^{\infty} (G)$ is the weak-* Haagerup tensor product $L^{\infty} (G) \otimes_{w * h} L^{\infty} (G)$ ), define the concept of X̂-operator synthesis and prove that a...

Metastability in reversible diffusion processes II: precise asymptotics for small eigenvalues

Anton Bovier, Véronique Gayrard, Markus Klein (2005)

Journal of the European Mathematical Society

Similarity:

We continue the analysis of the problem of metastability for reversible diffusion processes, initiated in [BEGK3], with a precise analysis of the low-lying spectrum of the generator. Recall that we are considering processes with generators of the form $- ϵ Δ + \nabla F (\cdot) \nabla$ on $ℝ^{d}$ or subsets of $ℝ^{d}$ , where $F$ is a smooth function with finitely many local minima. Here we consider only the generic situation where the depths of all local minima are different. We show that in general the exponentially small part of...

On polynomial robustness of flux reconstructions

Miloslav Vlasák (2020)

Applications of Mathematics

Similarity:

We deal with the numerical solution of elliptic not necessarily self-adjoint problems. We derive a posteriori upper bound based on the flux reconstruction that can be directly and cheaply evaluated from the original fluxes and we show for one-dimensional problems that local efficiency of the resulting a posteriori error estimators depends on $p^{1 / 2}$ only, where $p$ is the discretization polynomial degree. The theoretical results are verified by numerical experiments.

A spectral gap theorem in SU $(d)$

Jean Bourgain, Alex Gamburd (2012)

Journal of the European Mathematical Society

Similarity:

We establish the spectral gap property for dense subgroups of SU $(d)$ $(d \geq 2)$ , generated by finitely many elements with algebraic entries; this result was announced in [BG3]. The method of proof differs, in several crucial aspects, from that used in [BG] in the case of SU $(2)$ .

Numerical stability of the intrinsic equations for beams in time domain

Klesa, Jan

Similarity:

Intrinsic equations represent promising approach for the description of rotor blade dynamics. They are the system of non-linear partial differential equations. Stability of numeric solution by the finite difference method is described. The stability is studied for various numerical schemes with different methods for the computation of spatial derivatives from time level $n + 0.5$ (i.e., mean values of old and new time step) to $n + 1$ (i.e., only from new time step). Stable solution was obtained only...

On the best observation of wave and Schrödinger equations in quantum ergodic billiards

Yannick Privat, Emmanuel Trélat, Enrique Zuazua (2012)

Journées Équations aux dérivées partielles

Similarity:

This paper is a proceedings version of the ongoing work [20], and has been the object of the talk of the second author at Journées EDP in 2012. In this work we investigate optimal observability properties for wave and Schrödinger equations considered in a bounded open set $Ω \subset ℝ^{n}$ , with Dirichlet boundary conditions. The observation is done on a subset $ω$ of Lebesgue measure $| ω | = L | Ω |$ , where $L \in (0, 1)$ is fixed. We denote...