اشترك بالحزمة الذهبية واحصل على وصول غير محدود شمرا أكاديميا
تسجيل مستخدم جديدA posteriori error analysis for finite element solution of elliptic differential equations using equidistributing meshes
136
0
0.0
(
0
)
اسأل ChatGPT حول البحث
ﻻ يوجد ملخص باللغة العربية
The paper is concerned with the adaptive finite element solution of linear elliptic differential equations using equidistributing meshes. A strategy is developed for defining this type of mesh based on residual-based a posteriori error estimates and rigorously analyzing the convergence of a linear finite element approximation using them. The existence and computation of equidistributing meshes and the continuous dependence of the finite element approximation on mesh are also studied. Numerical results are given to verify the theoretical findings.
قيم البحث
اقرأ أيضاً
For the Stokes equation over 2D and 3D domains, explicit a posteriori and a priori error estimation are novelly developed for the finite element solution. The difficulty in handling the divergence-free condition of the Stokes equation is solved by ut
ilizing the extended hypercircle method along with the Scott-Vogelius finite element scheme. Since all terms in the error estimation have explicit values, by further applying the interval arithmetic and verified computing algorithms, the computed results provide rigorous estimation for the approximation error. As an application of the proposed error estimation, the eigenvalue problem of the Stokes operator is considered and rigorous bounds for the eigenvalues are obtained. The efficiency of proposed error estimation is demonstrated by solving the Stokes equation on both convex and non-convex 3D domains.
In this paper, we define new unfitted finite element methods for numerically approximating the solution of surface partial differential equations using bulk finite elements. The key idea is that the $n$-dimensional hypersurface, $Gamma subset mathbb{
R}^{n+1}$, is embedded in a polyhedral domain in $mathbb R^{n+1}$ consisting of a union, $mathcal{T}_h$, of $(n+1)$-simplices. The finite element approximating space is based on continuous piece-wise linear finite element functions on $mathcal{T}_h$. Our first method is a sharp interface method, emph{SIF}, which uses the bulk finite element space in an approximating weak formulation obtained from integration on a polygonal approximation, $Gamma_{h}$, of $Gamma$. The full gradient is used rather than the projected tangential gradient and it is this which distinguishes emph{SIF} from the method of [42]. The second method, emph{NBM}, is a narrow band method in which the region of integration is a narrow band of width $O(h)$. emph{NBM} is similar to the method of [13]. but again the full gradient is used in the discrete weak formulation. The a priori error analysis in this paper shows that the methods are of optimal order in the surface $L^{2}$ and $H^{1}$ norms and have the advantage that the normal derivative of the discrete solution is small and converges to zero. Our third method combines bulk finite elements, discrete sharp interfaces and narrow bands in order to give an unfitted finite element method for parabolic equations on evolving surfaces. We show that our method is conservative so that it preserves mass in the case of an advection diffusion conservation law. Numerical results are given which illustrate the rates of convergence.
Many practical problems occur due to the boundary value problem. This paper evaluates the finite element solution of the boundary value problem of Poissons equation and proposes a novel a posteriori local error estimation based on the Hypercircle met
hod. Compared to the existing literature on qualitative error estimation, the proposed error estimation provides an explicit and sharp bound for the approximation error in the subdomain of interest and is applicable to problems without the $H^2$ regularity. The efficiency of the proposed method is demonstrated by numerical experiments for both convex and non-convex 2D domains.
In this work we study a residual based a posteriori error estimation for the CutFEM method applied to an elliptic model problem. We consider the problem with non-polygonal boundary and the analysis takes into account the geometry and data approximati
on on the boundary. The reliability and efficiency are theoretically proved. Moreover, constants are robust with respect to how the domain boundary cuts the mesh.
Finite element exterior calculus (FEEC) has been developed over the past decade as a framework for constructing and analyzing stable and accurate numerical methods for partial differential equations by employing differential complexes. The recent wor
k of Arnold, Falk and Winther cite{ArFaWi2010} includes a well-developed theory of finite element methods for Hodge Laplace problems, including a priori error estimates. In this work we focus on developing a posteriori error estimates in which the computational error is bounded by some computable functional of the discrete solution and problem data. More precisely, we prove a posteriori error estimates of residual type for Arnold-Falk-Winther mixed finite element methods for Hodge-de Rham Laplace problems. While a number of previous works consider a posteriori error estimation for Maxwells equations and mixed formulations of the scalar Laplacian, the approach we take is distinguished by unified treatment of the various Hodge Laplace problems arising in the de Rham complex, consistent use of the language and analytical framework of differential forms, and the development of a posteriori error estimates for harmonic forms and the effects of their approximation on the resulting numerical method for the Hodge Laplacian.
سجل دخول لتتمكن من نشر تعليقات
التعليقات
جاري جلب التعليقات
سجل دخول لتتمكن من متابعة معايير البحث التي قمت باختيارها
