اشترك بالحزمة الذهبية واحصل على وصول غير محدود شمرا أكاديميا
تسجيل مستخدم جديدConstructive a priori error estimates for a full discrete approximation of periodic solutions for the heat equation
73
0
0.0
(
0
)
اسأل ChatGPT حول البحث
ﻻ يوجد ملخص باللغة العربية
We consider the constructive a priori error estimates for a full discrete numerical solution of the heat equation with time-periodic condition.
قيم البحث
اقرأ أيضاً
In this paper, we present a new full-discrete finite element method for the heat equation, and show the numerical stability of the method by verified computations. Since, in the error analysis, we use the constructive error estimates proposed ny Naka
o et. all in 2013, this work is considered as an extention of that paper. We emphasize that concerned scheme seems to be a quite normal Galerkin method and easy to implement for evolutionary equations comparing with previous one. In the constructive error estimates, we effectively use the numerical computations with guaranteed accuracy.
We present the error analysis of Lagrange interpolation on triangles. A new textit{a priori} error estimate is derived in which the bound is expressed in terms of the diameter and circumradius of a triangle. No geometric conditions on triangles are i
mposed in order to get this type of error estimates.
The Richards equation is commonly used to model the flow of water and air through soil, and it serves as a gateway equation for multiphase flows through porous media. It is a nonlinear advection-reaction-diffusion equation that exhibits both paraboli
c-hyperbolic and parabolic-elliptic kinds of degeneracies. In this study, we provide reliable, fully computable, and locally space-time efficient a posteriori error bounds for numerical approximations of the fully degenerate Richards equation. For showing global reliability, a nonlocal-in-time error estimate is derived individually for the time-integrated $H^1(H^{-1})$, $L^2(L^2)$, and the $L^2(H^1)$ errors. A maximum principle and a degeneracy estimator are employed for the last one. Global and local space-time efficiency error bounds are then obtained in a standard $H^1(H^{-1})cap L^2(H^1)$ norm. The reliability and efficiency norms employed coincide when there is no nonlinearity. Moreover, error contributors such as flux nonconformity, time discretization, quadrature, linearization, and data oscillation are identified and separated. The estimates are also valid in a setting where iterative linearization with inexact solvers is considered. Numerical tests are conducted for nondegenerate and degenerate cases having exact solutions, as well as for a realistic case. It is shown that the estimators correctly identify the errors up to a factor of the order of unity.
Elliptic partial differential equations on surfaces play an essential role in geometry, relativity theory, phase transitions, materials science, image processing, and other applications. They are typically governed by the Laplace-Beltrami operator. W
e present and analyze approximations by Surface Finite Element Methods (SFEM) of the Laplace-Beltrami eigenvalue problem. As for SFEM for source problems, spectral approximation is challenged by two sources of errors: the geometric consistency error due to the approximation of the surface and the Galerkin error corresponding to finite element resolution of eigenfunctions. We show that these two error sources interact for eigenfunction approximations as for the source problem. The situation is different for eigenvalues, where a novel situation occurs for the geometric consistency error: The degree of the geometric error depends on the choice of interpolation points used to construct the approximate surface. Thus the geometric consistency term can sometimes be made to converge faster than in the eigenfunction case through a judicious choice of interpolation points.
The logarithmic nonlinearity has been used in many partial differential equations (PDEs) for modeling problems in various applications.Due to the singularity of the logarithmic function, it introducestremendous difficulties in establishing mathematic
al theories, as well asin designing and analyzing numerical methods for PDEs with such nonlinearity. Here we take the logarithmic Schrodinger equation (LogSE)as a prototype model. Instead of regularizing $f(rho)=ln rho$ in theLogSE directly and globally as being done in the literature, we propose a local energy regularization (LER) for the LogSE byfirst regularizing $F(rho)=rholn rho -rho$ locally near $rho=0^+$ with a polynomial approximation in the energy functional of the LogSE and then obtaining an energy regularized logarithmic Schrodinger equation (ERLogSE) via energy variation. Linear convergence is established between the solutions of ERLogSE and LogSE in terms of a small regularization parameter $0<epll1$. Moreover, the conserved energy of the ERLogSE converges to that of LogSE quadratically, which significantly improvesthe linear convergence rate of the regularization method in the literature. Error estimates are alsopresented for solving the ERLogSE by using Lie-Trotter splittingintegrators. Numerical results are reported to confirm our errorestimates of the LER and of the time-splitting integrators for theERLogSE. Finally our results suggest that the LER performs better than regularizing the logarithmic nonlinearity in the LogSE directly.
سجل دخول لتتمكن من نشر تعليقات
التعليقات
جاري جلب التعليقات
سجل دخول لتتمكن من متابعة معايير البحث التي قمت باختيارها
