On polynomial robustness of flux reconstructions

Miloslav Vlasák

Displaying similar documents to “On polynomial robustness of flux reconstructions”

The gradient superconvergence of the finite volume method for a nonlinear elliptic problem of nonmonotone type

Tie Zhu Zhang, Shu Hua Zhang (2015)

Applications of Mathematics

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We study the superconvergence of the finite volume method for a nonlinear elliptic problem using linear trial functions. Under the condition of $C$ -uniform meshes, we first establish a superclose weak estimate for the bilinear form of the finite volume method. Then, we prove that on the mesh point set $S$ , the gradient approximation possesses the superconvergence: ${max}_{P \in S} | (\nabla u - \bar{\nabla} u_{h}) (P) | = O (h^{2}) {| ln h |}^{3 / 2}$ , where $\bar{\nabla}$ denotes the average gradient on elements containing vertex $P$ . Furthermore, by using the interpolation post-processing...

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

Olivier Ramaré (2013)

Acta Arithmetica

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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.

Uniform $L^{1}$ error bounds for semi-discrete finite element solutions of evolutionary integral equations

Lin, Qun, Xu, Da, Zhang, Shuhua

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In this paper, we consider the second-order continuous time Galerkin approximation of the solution to the initial problem $u_{t} + \int_{0}^{t} β (t - s) A u (s) d s = 0, u (0) = v, t > 0,$ where A is an elliptic partial-differential operator and $β (t)$ is positive, nonincreasing and log-convex on $(0, \infty)$ with $0 \leq β (\infty) < β (0^{+}) \leq \infty$ . Error estimates are derived in the norm of $L_{t}^{1} (0, \infty; L_{x}^{2})$ , and some estimates for the first order time derivatives of the errors are also given.

Finite element analysis for a regularized variational inequality of the second kind

Zhang, Tie, Zhang, Shuhua, Azari, Hossein

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In this paper, we investigate the a priori and the a posteriori error analysis for the finite element approximation to a regularization version of the variational inequality of the second kind. We prove the abstract optimal error estimates in the $H^{1}$ - and $L_{2}$ -norms, respectively, and also derive the optimal order error estimate in the $L_{\infty}$ -norm under the strongly regular triangulation condition. Moreover, some residual–based a posteriori error estimators are established, which can provide the...

Anisotropic $h p$ -adaptive method based on interpolation error estimates in the $H^{1}$ -seminorm

Vít Dolejší (2015)

Applications of Mathematics

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We develop a new technique which, for the given smooth function, generates the anisotropic triangular grid and the corresponding polynomial approximation degrees based on the minimization of the interpolation error in the broken $H^{1}$ -seminorm. This technique can be employed for the numerical solution of boundary value problems with the aid of finite element methods. We present the theoretical background of this approach and show several numerical examples demonstrating the efficiency of...

Space-time adaptive $h p$ -FEM: Methodology overview

Šolín, Pavel, Segeth, Karel, Doležel, Ivo

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We present a new class of self-adaptive higher-order finite element methods ( $h p$ -FEM) which are free of analytical error estimates and thus work equally well for virtually all PDE problems ranging from simple linear elliptic equations to complex time-dependent nonlinear multiphysics coupled problems. The methods do not contain any tuning parameters and work reliably with both low- and high-order finite elements. The methodology was used to solve various types of problems including thermoelasticity,...

Partially elliptic differential equations having distributions as their right members

H. Marcinkowska

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ContentsIntroduction.............................................................................................................................31. Definitions, notations and some auxiliary lemmas...................................................42. The definition of the spaces $H_{p, q; Y} (Ω, ℬ)$ ..........................................................73. Some properties of the spaces $H_{p, q; Y} (Ω, ℬ)$ ...................................................104. Some examples of the spaces $H_{p, q; Y} (Ω, ℬ)$ ....................................................155....

Explicit estimation of error constants appearing in non-conforming linear triangular finite element method

Xuefeng Liu, Fumio Kikuchi (2018)

Applications of Mathematics

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The non-conforming linear ( $P_{1}$ ) triangular FEM can be viewed as a kind of the discontinuous Galerkin method, and is attractive in both the theoretical and practical purposes. Since various error constants must be quantitatively evaluated for its accurate a priori and a posteriori error estimates, we derive their theoretical upper bounds and some computational results. In particular, the Babuška-Aziz maximum angle condition is required just as in the case of the conforming $P_{1}$ triangle. Some...

Error analysis of splitting methods for semilinear evolution equations

Masahito Ohta, Takiko Sasaki (2017)

Applications of Mathematics

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We consider a Strang-type splitting method for an abstract semilinear evolution equation $\partial_{t} u = A u + F (u) .$ Roughly speaking, the splitting method is a time-discretization approximation based on the decomposition of the operators $A$ and $F .$ Particularly, the Strang method is a popular splitting method and is known to be convergent at a second order rate for some particular ODEs and PDEs. Moreover, such estimates usually address the case of splitting the operator into two parts. In this paper, we consider...

Unified error analysis of discontinuous Galerkin methods for parabolic obstacle problem

Papri Majumder (2021)

Applications of Mathematics

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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...