اشترك بالحزمة الذهبية واحصل على وصول غير محدود شمرا أكاديميا
تسجيل مستخدم جديدWeighted error estimates for transient transport problems discretized using continuous finite elements with interior penalty stabilization on the gradient jumps
141
0
0.0
(
0
)
تأليف
Erik Burman
اسأل ChatGPT حول البحث
ﻻ يوجد ملخص باللغة العربية
In this paper we consider the semi-discretization in space of a first order scalar transport equation. For the space discretization we use standard continuous finite elements. To obtain stability we add a penalty on the jump of the gradient over element faces. We recall some global error estimates for smooth and rough solutions and then prove a new local error estimate for the transient linear transport equation. In particular we show that in the stabilized method the effect of non-smooth features in the solution decay exponentially from the space time zone where the solution is rough so that smooth features will be transported unperturbed. Locally the $L^2$-norm error converges with the expected order $O(h^{k+frac12})$. We then illustrate the results numerically. In particular we show the good local accuracy in the smooth zone of the stabilized method and that the standard Galerkin fails to approximate a solution that is smooth at the final time if discontinuities have been present in the solution at some time during the evolution.
قيم البحث
اقرأ أيضاً
We consider a finite element method with symmetric stabilisation for the discretisation of the transient convection--diffusion equation. For the time-discretisation we consider either the second order backwards differentiation formula or the Crank-Ni
colson method. Both the convection term and the associated stabilisation are treated explicitly using an extrapolated approximate solution. We prove stability of the method and the $tau^2 + h^{p+{frac12}}$ error estimates for the $L^2$-norm under either the standard hyperbolic CFL condition, when piecewise affine ($p=1$) approximation is used, or in the case of finite element approximation of order $p ge 1$, a stronger, so-called $4/3$-CFL, i.e. $tau leq C h^{4/3}$. The theory is illustrated with some numerical examples.
We propose a family of mixed finite element that is robust for the nearly incompressible strain gradient model, which is a fourth order singular perturbation elliptic system. The element is similar to the Taylor-Hood element in the Stokes flow. Using
a uniform stable Fortin operator for the mixed finite element pairs, we are able to prove the optimal rate of convergence that is robust in the incompressible limit. Moreover, we estimate the convergence rate of the numerical solution to the unperturbed second order elliptic system. Numerical results for both smooth solutions and the solutions with sharp layers confirm the theoretical prediction.
In this paper, the optimal choice of the interior penalty parameter of the discontinuous Galerkin finite element methods for both the elliptic problems and the Biots systems are studied by utilizing the neural network and machine learning. It is cruc
ial to choose the optimal interior penalty parameter, which is not too small or not too large for the stability, robustness, and efficiency of the numerical discretized solutions. Both linear regression and nonlinear artificial neural network methods are employed and compared using several numerical experiments to illustrate the capability of our proposed computational framework. This framework is an integral part of a developing automated numerical simulation platform because it can automatically identify the optimal interior penalty parameter. Real-time feedback could also be implemented to update and improve model accuracy on the fly.
In the present work we show some results on the effect of the Smagorinsky model on the stability of the associated perturbation equation. We show that in the presence of a spectral gap, such that the flow can be decomposed in a large scale with moder
ate gradient and a small amplitude fine scale with arbitratry gradient, the Smagorinsky model admits stability estimates for perturbations, with exponential growth depending only on the large scale gradient. We then show in the context of stabilized finite element methods that the same result carries over to the approximation and that in this context, for suitably chosen finite element spaces the Smagorinsky model acts as a stabilizer yielding close to optimal error estimates in the $L^2$-norm for smooth flows in the pre-asymptotic high Reynolds number regime.
A general adaptive refinement strategy for solving linear elliptic partial differential equation with random data is proposed and analysed herein. The adaptive strategy extends the a posteriori error estimation framework introduced by Guignard and No
bile in 2018 (SIAM J. Numer. Anal., 56, 3121--3143) to cover problems with a nonaffine parametric coefficient dependence. A suboptimal, but nonetheless reliable and convenient implementation of the strategy involves approximation of the decoupled PDE problems with a common finite element approximation space. Computational results obtained using such a single-level strategy are presented in this paper (part I). Results obtained using a potentially more efficient multilevel approximation strategy, where meshes are individually tailored, will be discussed in part II of this work. The codes used to generate the numerical results are available online.
سجل دخول لتتمكن من نشر تعليقات
التعليقات
جاري جلب التعليقات
سجل دخول لتتمكن من متابعة معايير البحث التي قمت باختيارها
