scholarly journals A Polygonal Finite Element Method for Stokes Equations

2021 ◽  
pp. 62-78
Author(s):  
Xinjiang Chen
Keyword(s):  
Finite Element ◽  
Convergence Rates ◽  
Stokes Equations ◽  
Mixed Finite Element ◽  
Barycentric Coordinates ◽  
Polygonal Meshes ◽  
Lbb Condition ◽  
Optimal Convergence Rates ◽  
Generalized Barycentric Coordinates ◽  
Element Theory

In this paper, we extend the Bernardi-Raugel element [1] to convex polygonal meshes by using the generalized barycentric coordinates. Comparing to traditional discretizations defined on triangular and rectangular meshes, polygonal meshes can be more flexible when dealing with complicated domains or domains with curved boundaries. Theoretical analysis of the new element follows the standard mixed finite element theory for Stokes equations, i.e., we shall prove the discrete inf-sup condition (LBB condition) by constructing a Fortin operator. Because there is no scaling argument on polygonal meshes and the generalized barycentric coordinates are in general not polynomials, special treatments are required in the analysis. We prove that the extended Bernardi-Raugel element has optimal convergence rates. Supporting numerical results are also presented. 

A NEW NONCONFORMING MIXED FINITE ELEMENT METHOD FOR LINEAR ELASTICITY

2006 ◽  
Vol 16 (07) ◽  
pp. 979-999 ◽  
Author(s):  
SON-YOUNG YI
Keyword(s):  
Finite Element ◽  
Linear Elasticity ◽  
Convergence Rates ◽  
Mixed Finite Element ◽  
Mixed Finite Element Method ◽  
Optimal Order ◽  
Displacement Boundary Condition ◽  
Displacement Boundary ◽  
Optimal Convergence Rates ◽  
Optimal Order Convergence

We have developed new nonconforming mixed finite element methods for linear elasticity with a pure traction (displacement) boundary condition based on the Hellinger–Reissner variational principle using rectangular elements. Convergence analysis yields an optimal (suboptimal) convergence rate of [Formula: see text] for the L2-error of the stress and [Formula: see text] for the displacement in the pure traction (displacement) boundary problem. However, numerical experiments have yielded optimal-order convergence rates for both stress and displacement in both problems and have shown superconvergence for the displacement at the midpoint of each element. Moreover, we observed that the optimal convergence rates are still valid for large λ.

A Banach spaces-based analysis of a new fully-mixed finite element method for the Boussinesq problem

2020 ◽  
Vol 54 (5) ◽  
pp. 1525-1568 ◽  
Author(s):  
Eligio Colmenares ◽  
Gabriel N. Gatica ◽  
Sebastián Moraga
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Banach Spaces ◽  
Stokes Equations ◽  
Mixed Finite Element ◽  
Mixed Finite Element Method ◽  
Small Data ◽  
Temperature Dependent Viscosity ◽  
Boussinesq Problem ◽  
Element Method

In this paper we propose and analyze, utilizing mainly tools and abstract results from Banach spaces rather than from Hilbert ones, a new fully-mixed finite element method for the stationary Boussinesq problem with temperature-dependent viscosity. More precisely, following an idea that has already been applied to the Navier–Stokes equations and to the fluid part only of our model of interest, we first incorporate the velocity gradient and the associated Bernoulli stress tensor as auxiliary unknowns. Additionally, and differently from earlier works in which either the primal or the classical dual-mixed method is employed for the heat equation, we consider here an analogue of the approach for the fluid, which consists of introducing as further variables the gradient of temperature and a vector version of the Bernoulli tensor. The resulting mixed variational formulation, which involves the aforementioned four unknowns together with the original variables given by the velocity and temperature of the fluid, is then reformulated as a fixed point equation. Next, we utilize the well-known Banach and Brouwer theorems, combined with the application of the Babuška-Brezzi theory to each independent equation, to prove, under suitable small data assumptions, the existence of a unique solution to the continuous scheme, and the existence of solution to the associated Galerkin system for a feasible choice of the corresponding finite element subspaces. Finally, we derive optimal a priori error estimates and provide several numerical results illustrating the performance of the fully-mixed scheme and confirming the theoretical rates of convergence.

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