Semidiscrete least squares methods for linear convection-diffusion problem (Q1203678)

The paper deals with the following time-dependent convection-dominated linear convection-diffusion problem: \(u_ t+\beta\cdot\nabla u- \varepsilon\Delta u=f\), \(x\in\Omega\), \(t\in(0,T)\); \(u(x,0)=u_ 0(x)\), \(x\in\Omega\); \(u(x,t)=0\), \(x\in\Gamma\), \(t\in(0,T)\), where \(\Omega\) is a bounded domain in \(\mathbb{R}^ 2\) with boundary \(\Gamma\), \(\beta=(\beta_ 1,\beta_ 2)\) is a constant vector and \(\varepsilon>0\) is a small constant. The problem is reduced to an elliptic one by using the Crank-Nicolson scheme. For the resulting problem, two versions of the least squares method are considered. An error analysis of these methods is given. Numerical results for the nonlinear Burgers equation \(\varphi_ t+\varphi\varphi_ x-\varepsilon\varphi_{xx}=0\) are also presented.