Coefficient Jump-Independent Approximation of the Conforming and Nonconforming Finite Element Solutions

2016 ◽  
Vol 8 (5) ◽  
pp. 722-736
Author(s):  
Shangyou Zhang
Keyword(s):  
Differential Equation ◽  
Finite Element ◽  
Elliptic Equations ◽  
Order Approximation ◽  
Elliptic Differential Equation ◽  
Optimal Order ◽  
Nonconforming Finite Element ◽  
Numerical Tests ◽  
Conforming Finite Element ◽  
Theoretical Finding

AbstractA counterexample is constructed. It confirms that the error of conforming finite element solution is proportional to the coefficient jump, when solving interface elliptic equations. The Scott-Zhang operator is applied to a nonconforming finite element. It is shown that the nonconforming finite element provides the optimal order approximation in interpolation, in L2-projection, and in solving elliptic differential equation, independent of the coefficient jump in the elliptic differential equation. Numerical tests confirm the theoretical finding.

scholarly journals L2-error analysis of an isoparametric unfitted finite element method for elliptic interface problems

2019 ◽  
Vol 27 (2) ◽  
pp. 85-99
Author(s):  
Christoph Lehrenfeld ◽  
Arnold Reusken
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Error Analysis ◽  
Error Bounds ◽  
Order Approximation ◽  
High Order ◽  
Optimal Order ◽  
Interface Problem ◽  
High Order Approximation ◽  
Element Method

AbstractIn the context of unfitted finite element discretizations the realization of high order methods is challenging due to the fact that the geometry approximation has to be sufficiently accurate. Recently a new unfitted finite element method was introduced which achieves a high order approximation of the geometry for domains which are implicitly described by smooth level set functions. This method is based on a parametric mapping which transforms a piecewise planar interface (or surface) reconstruction to a high order approximation. In the paper [C. Lehrenfeld and A. Reusken,IMA J. Numer. Anal.38(2018), No. 3, 1351–1387] ana priorierror analysis of the method applied to an interface problem is presented. The analysis reveals optimal order discretization error bounds in theH1-norm. In this paper we extend this analysis and derive optimalL2-error bounds.

scholarly journals Stabilized Multiscale Nonconforming Finite Element Method for the Stationary Navier-Stokes Equations

2012 ◽  
Vol 2012 ◽  
pp. 1-27
Author(s):  
Tong Zhang ◽  
Shunwei Xu ◽  
Jien Deng
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Stokes Equations ◽  
Stokes Problem ◽  
Navier Stokes ◽  
Optimal Order ◽  
Nonconforming Finite Element ◽  
Polynomial Space ◽  
Nonconforming Finite Element Method ◽  
Element Method

We consider a stabilized multiscale nonconforming finite element method for the two-dimensional stationary incompressible Navier-Stokes problem. This method is based on the enrichment of the standard polynomial space for the velocity component with multiscale function and the nonconforming lowest equal-order finite element pair. Stability and existence uniqueness of the numerical solution are established, optimal-order error estimates are also presented. Finally, some numerical results are presented to validate the performance of the proposed method.

scholarly journals Integral operator and oscillation of second order elliptic equations

2005 ◽  
Vol 36 (2) ◽  
pp. 93-101 ◽  
Author(s):  
Zhiting Xu ◽  
Hongyan Xing
Keyword(s):  
Differential Equation ◽  
Integral Operator ◽  
Elliptic Equations ◽  
Second Order ◽  
Elliptic Differential Equation ◽  
Oscillation Criteria ◽  
Second Order Elliptic Equations

By using integral operator, some oscillation criteria for second order elliptic differential equation$$ \sum^d _{i,j=1} D_i[A_{ij}(x)D_jy]+ q(x)f(y)=0, \;x \in \Omega\qquad \eqno{(E)} $$are established. The results obtained here can be regarded as the extension of the well-known Kamenev theorem to Eq.$(E)$.

Finite element analysis of the constrained Dirichlet boundary control problem governed by the diffusion problem

2020 ◽  
Vol 26 ◽  
pp. 78
Author(s):  
Thirupathi Gudi ◽  
Ramesh Ch. Sau
Keyword(s):  
Finite Element ◽  
Control Problem ◽  
Dirichlet Boundary ◽  
A Priori ◽  
Diffusion Problem ◽  
Discrete Control ◽  
Optimal Order ◽  
Control Constraints ◽  
Element Analysis ◽  
Dirichlet Boundary Control

We study an energy space-based approach for the Dirichlet boundary optimal control problem governed by the Laplace equation with control constraints. The optimality system results in a simplified Signorini type problem for control which is coupled with boundary value problems for state and costate variables. We propose a finite element based numerical method using the linear Lagrange finite element spaces with discrete control constraints at the Lagrange nodes. The analysis is presented in a combination for both the gradient and the L2 cost functional. A priori error estimates of optimal order in the energy norm is derived up to the regularity of the solution for both the cases. Theoretical results are illustrated by some numerical experiments.

Error analysis of higher order trace finite element methods for the surface Stokes equation

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Thomas Jankuhn ◽  
Maxim A. Olshanskii ◽  
Arnold Reusken ◽  
Alexander Zhiliakov
Keyword(s):  
Finite Element ◽  
Stokes System ◽  
Parametric Representation ◽  
Higher Order ◽  
Optimal Order ◽  
Helmholtz Instability ◽  
Surface Stability ◽  
Instability Problem ◽  
Convergence Results ◽  
Optimal Order Convergence

AbstractThe paper studies a higher order unfitted finite element method for the Stokes system posed on a surface in ℝ3. The method employs parametric Pk-Pk−1 finite element pairs on tetrahedral bulk mesh to discretize the Stokes system on embedded surface. Stability and optimal order convergence results are proved. The proofs include a complete quantification of geometric errors stemming from approximate parametric representation of the surface. Numerical experiments include formal convergence studies and an example of the Kelvin--Helmholtz instability problem on the unit sphere.

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