Scientific publications
* (Corresponding author)
Mahamat Saleh Daoussa Haggar
H. Oleï Tahar
Journal of Applied Mathematics
This paper is devoted to the study of numerical approximation for a class of two-dimensional Navier-Stokes equations with slip boundary conditions of friction type. The objective is to establish the well-posedness and stability of the numerical scheme in <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" id="M1"> <a:msup> <a:mrow> <a:mi>L</a:mi> </a:mrow> <a:mrow> <a:mn>2</a:mn> </a:mrow> </a:msup> </a:math> -norm and <c:math xmlns:c="http://www.w3.org/1998/Math/MathML" id="M2"> <c:msup> <c:mrow> <c:mi>H</c:mi> </c:mrow> <c:mrow> <c:mn>1</c:mn> </c:mrow> </c:msup> </c:math> -norm for all positive time using the Crank-Nicholson scheme in time and the finite element approximation in space. The resulting variational structure dealing with is in the form of inequality, and obtaining <e:math xmlns:e="http://www.w3.org/1998/Math/MathML" id="M3"> <e:msup> <e:mrow> <e:mi>H</e:mi> </e:mrow> <e:mrow> <e:mn>1</e:mn> </e:mrow> </e:msup> </e:math> -estimate is more involved because of the presence of the nondifferentiable term appearing at the boundary where slip occurs. We prove that the numerical scheme is stable in <g:math xmlns:g="http://www.w3.org/1998/Math/MathML" id="M4"> <g:msup> <g:mrow> <g:mi>L</g:mi> </g:mrow> <g:mrow> <g:mn>2</g:mn> </g:mrow> </g:msup> </g:math> and <i:math xmlns:i="http://www.w3.org/1998/Math/MathML" id="M5"> <i:msup> <i:mrow> <i:mi>H</i:mi> </i:mrow> <i:mrow> <i:mn>1</i:mn> </i:mrow> </i:msup> </i:math> -norms with the aid of different versions of discrete Grownwall lemmas, under a CFL-type condition. Finally, some numerical simulations are presented to illustrate our theoretical analysis.
