A Posteriori Error Analysis for the Weak Galerkin Method for Solving Elliptic Problems

2018 ◽  
Vol 15 (08) ◽  
pp. 1850075 ◽  
Author(s):  
Tie Zhang ◽  
Yanli Chen
Keyword(s):  
Elliptic Problems ◽  
Posteriori Error ◽  
Upper And Lower Bounds ◽  
Posteriori Error Estimate ◽  
Error Estimator ◽  
A Posteriori ◽  
Weak Galerkin ◽  
A Posteriori Error ◽  
Galerkin Finite Element ◽  
Posteriori Error Analysis

In this paper, we study the a posteriori error estimate for weak Galerkin finite element method solving elliptic problems. A residual type error estimator is proposed and is proven to be reliable and efficient. This estimator provides global upper and lower bounds on the exact error in a discrete [Formula: see text]-norm. Numerical experiments are given to illustrate the effectiveness of the proposed error estimator.

scholarly journals Residual-based a posteriori error analysis for the coupling of the Navier-Stokes and Darcy-Forchheimer equations

2021 ◽  
Author(s):  
Sergio Caucao ◽  
Gabriel Gatica ◽  
Ricardo Oyarzúa ◽  
Felipe Sandoval
Keyword(s):  
Adaptive Mesh Refinement ◽  
Adaptive Mesh ◽  
Posteriori Error ◽  
Local Approximation ◽  
Navier Stokes ◽  
Error Estimator ◽  
A Posteriori ◽  
Approximation Properties ◽  
A Posteriori Error ◽  
Posteriori Error Analysis

In this paper we consider a mixed variational formulation that have been recently proposed for the coupling of the Navier--Stokes and Darcy--Forchheimer equations, and derive,  though in a non-standard sense,  a reliable and efficient residual-based a posteriori error estimator suitable for an adaptive mesh-refinement method.  For the reliability estimate, which holds with respect to the square root of the error estimator, we make use of the inf-sup condition and the strict monotonicity of the operators involved, a suitable Helmholtz decomposition in non-standard Banach spaces in the porous medium, local approximation properties of the Cl\'ement interpolant and Raviart--Thomas operator, and a smallness assumption on the data.   In turn, inverse inequalities, the localization technique based on triangle-bubble and edge-bubble functions in local $\L^\rp$ spaces, are the main tools for developing the effi\-ciency analysis, which is valid for the error estimator itself up to a suitable additional error term. Finally, several numerical results confirming the properties of the estimator and illustrating the performance of the associated adaptive algorithm are reported.

A Posteriori Error Estimator for a Weak Galerkin Finite Element Solution of the Stokes Problem

2017 ◽  
Vol 7 (3) ◽  
pp. 508-529 ◽  
Author(s):  
Xiaobo Zheng ◽  
Xiaoping Xie
Keyword(s):  
Finite Element ◽  
Lower Bounds ◽  
Stokes Problem ◽  
Three Dimensions ◽  
Upper And Lower Bounds ◽  
Error Estimator ◽  
A Posteriori ◽  
Weak Galerkin ◽  
Galerkin Finite Element ◽  
Velocity Approximation

AbstractA robust residual-based a posteriori error estimator is proposed for a weak Galerkin finite element method for the Stokes problem in two and three dimensions. The estimator consists of two terms, where the first term characterises the difference between the L2-projection of the velocity approximation on the element interfaces and the corresponding numerical trace, and the second is related to the jump of the velocity approximation between the adjacent elements. We show that the estimator is reliable and efficient through two estimates of global upper and global lower bounds, up to two data oscillation terms caused by the source term and the nonhomogeneous Dirichlet boundary condition. The estimator is also robust in the sense that the constant factors in the upper and lower bounds are independent of the viscosity coefficient. Numerical results are provided to verify the theoretical results.

scholarly journals A Posteriori Error Analysis for a Lagrange Multiplier Method for a Stokes/Biot Fluid-Poroelastic Structure Interaction Model

2021 ◽  
Vol 2021 ◽  
pp. 1-12
Author(s):  
Houédanou Koffi Wilfrid
Keyword(s):  
Finite Element ◽  
Error Analysis ◽  
Error Estimate ◽  
Posteriori Error ◽  
A Posteriori Error Estimate ◽  
Posteriori Error Estimate ◽  
A Posteriori ◽  
A Posteriori Error Analysis ◽  
A Posteriori Error ◽  
Posteriori Error Analysis

In this work, we develop an a posteriori error analysis of a conforming mixed finite element method for solving the coupled problem arising in the interaction between a free fluid and a fluid in a poroelastic medium on isotropic meshes in ℝ d , d ∈ 2 , 3 . The approach utilizes a Lagrange multiplier method to impose weakly the interface conditions [Ilona Ambartsumyan et al., Numerische Mathematik, 140 (2): 513-553, 2018]. The a posteriori error estimate is based on a suitable evaluation on the residual of the finite element solution. It is proven that the a posteriori error estimate provided in this paper is both reliable and efficient. The proof of reliability makes use of suitable auxiliary problems, diverse continuous inf-sup conditions satisfied by the bilinear forms involved, Helmholtz decomposition, and local approximation properties of the Clément interpolant. On the other hand, inverse inequalities and the localization technique based on simplexe-bubble and face-bubble functions are the main tools for proving the efficiency of the estimator. Up to minor modifications, our analysis can be extended to other finite element subspaces yielding a stable Galerkin scheme.

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