Stabilised hp -Finite Element Approximation of Partial Differential Equations with Nonnegative Characteristic Form

Computing ◽  
2001 ◽  
Vol 66 (2) ◽  
pp. 99-119 ◽  
Author(s):  
P. Houston ◽  
Endre Süli
Keyword(s):  
Finite Element ◽  
Partial Differential Equations ◽  
Differential Equations ◽  
Finite Element Approximation ◽  
Element Approximation ◽  
Characteristic Form ◽  
Partial Differential ◽  
Nonnegative Characteristic Form ◽  
Hp Finite Element

scholarly journals Finite element approximation of an incompressible chemically reacting non-Newtonian fluid

2018 ◽  
Vol 52 (2) ◽  
pp. 509-541 ◽  
Author(s):  
Seungchan Ko ◽  
Petra Pustějovská ◽  
Endre Süli
Keyword(s):  
Finite Element ◽  
Partial Differential Equations ◽  
Differential Equations ◽  
Newtonian Fluid ◽  
Power Law ◽  
Nonlinear Partial Differential Equations ◽  
Finite Element Approximation ◽  
Element Approximation ◽  
Partial Differential ◽  
Chemically Reacting

We consider a system of nonlinear partial differential equations modelling the steady motion of an incompressible non-Newtonian fluid, which is chemically reacting. The governing system consists of a steady convection-diffusion equation for the concentration and the generalized steady Navier–Stokes equations, where the viscosity coefficient is a power-law type function of the shear-rate, and the coupling between the equations results from the concentration-dependence of the power-law index. This system of nonlinear partial differential equations arises in mathematical models of the synovial fluid found in the cavities of moving joints. We construct a finite element approximation of the model and perform the mathematical analysis of the numerical method in the case of two space dimensions. Key technical tools include discrete counterparts of the Bogovskiĭ operator, De Giorgi’s regularity theorem in two dimensions, and the Acerbi–Fusco Lipschitz truncation of Sobolev functions, in function spaces with variable integrability exponents.

scholarly journals Finite element approximation of steady flows of colloidal solutions

2021 ◽  
Author(s):  
Andrea Bonito ◽  
Vivette Girault ◽  
Diane Guignard ◽  
Kumbakonam R. Rajagopal ◽  
Endre Süli
Keyword(s):  
Finite Element ◽  
Weak Solution ◽  
Partial Differential Equations ◽  
Differential Equations ◽  
Nonlinear Partial Differential Equations ◽  
Finite Element Approximation ◽  
Element Approximation ◽  
Cauchy Stress ◽  
Partial Differential ◽  
Fixed Point Algorithm

We consider the mathematical analysis and numerical approximation of a system of nonlinear partial differential equations that arises in models that have relevance to  steady isochoric flows of colloidal suspensions. The symmetric velocity gradient is assumed to be a monotone nonlinear function of the deviatoric part of the Cauchy stress tensor. We prove the existence of a weak solution to the problem, and under the additional assumption that the nonlinearity involved in the constitutive relation is Lipschitz continuous we also prove uniqueness of the weak solution. We then construct mixed finite element approximations of the system using both conforming and nonconforming finite element spaces. For both of these we prove the convergence of the method to the unique weak solution of the problem, and in the case of the conforming method we provide a bound on the error between the analytical solution and its finite element approximation in terms of the best approximation error from the finite element spaces. We propose first a Lions-Mercier type iterative method and next a classical fixed-point algorithm to solve the finite-dimensional problems resulting from the finite element discretisation of the system of nonlinear partial differential equations under consideration and present numerical experiments that illustrate the practical performance of the proposed numerical method.

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