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.

scholarly journals A Posteriori Error Analysis for a New Fully Mixed Isotropic Discretization of the Stationary Stokes-Darcy Coupled Problem

2020 ◽  
Vol 2020 ◽  
pp. 1-12
Author(s):  
Houédanou Koffi Wilfrid ◽  
Adetola Jamal ◽  
Allaoui Mohamed
Keyword(s):  
Finite Element ◽  
Error Estimate ◽  
Posteriori Error ◽  
A Posteriori Error Estimate ◽  
Posteriori Error Estimate ◽  
A Posteriori ◽  
A Posteriori Error Analysis ◽  
A Posteriori Error ◽  
Coupled Problem ◽  
Posteriori Error Analysis

In this paper, we develop an a posteriori error analysis for the stationary Stokes-Darcy coupled problem approximated by conforming the finite element method on isotropic meshes in ℝ d , d ∈ 2 , 3 . The approach utilizes a new robust stabilized fully mixed discretization. The a posteriori error estimate is based on a suitable evaluation on the residual of the finite element solution plus the stabilization terms. It is proven that the a posteriori error estimate provided in this paper is both reliable and efficient.

Unified a Priori Error Estimate and a Posteriori Error Estimate of CIP-FEM for Elliptic Equations

2016 ◽  
Vol 8 (4) ◽  
pp. 517-535
Author(s):  
Jianye Wang ◽  
Rui Ma
Keyword(s):  
Finite Element ◽  
Error Analysis ◽  
Error Estimate ◽  
A Priori ◽  
Posteriori Error ◽  
A Posteriori ◽  
A Posteriori Error Analysis ◽  
A Posteriori Error ◽  
Penalty Term ◽  
Posteriori Error Analysis

AbstractThis paper is devoted to a unified a priori and a posteriori error analysis of CIP-FEM (continuous interior penalty finite element method) for second-order elliptic problems. Compared with the classic a priori error analysis in literature, our technique can easily apply for any type regularity assumption on the exact solution, especially for the case of lowerH1+sweak regularity under consideration, where 0 ≤s≤ 1/2. Because of the penalty term used in the CIP-FEM, Galerkin orthogonality is lost and Céa Lemma for conforming finite element methods can not be applied immediately when 0≤s≤1/2. To overcome this difficulty, our main idea is introducing an auxiliaryC1finite element space in the analysis of the penalty term. The same tool is also utilized in the explicit a posteriori error analysis of CIP-FEM.

scholarly journals An adaptive finite element DtN method for the elastic wave scattering by biperiodic structures

2021 ◽  
Author(s):  
Gang Bao ◽  
Xue Jiang ◽  
Peijun Li ◽  
Xiaokai Yuan
Keyword(s):  
Finite Element ◽  
Error Estimate ◽  
Elastic Wave ◽  
Posteriori Error ◽  
A Posteriori Error Estimate ◽  
Posteriori Error Estimate ◽  
Adaptive Finite Element ◽  
A Posteriori ◽  
Truncation Parameter ◽  
A Posteriori Error

Consider the scattering of a time-harmonic elastic plane wave by a bi-periodic rigid surface. The displacement of elastic wave motion is modeled by the three-dimensional Navier equation in an unbounded domain above the surface. Based on the Dirichlet-to-Neumann (DtN) operator, which is given as an infinite series, an exact transparent boundary condition is introduced and the scattering problem is formulated equivalently into a boundary value problem in a bounded domain. An a posteriori error estimate based adaptive finite element DtN method is proposed to solve the discrete variational problem where the DtN operator is truncated into a finite number of terms. The a posteriori error estimate takes account of the finite element approximation error and the truncation error of the DtN operator which is shown to decay exponentially with respect to the truncation parameter. Numerical experiments are presented to illustrate the effectiveness of the proposed method.

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