Uniform convergent monotone iterates for semilinear singularly perturbed parabolic problems (Q534237)

The author constructs a nonlinear monotone iterative scheme for solving semilinear singularly perturbed parabolic problems of the form \(u_t-Lu+f(x,y,t,u)=0\) for \((x,y,t)\in\omega\times(0,T]\), with boundary conditions \(u(x,y,t)=g(x,y,t)\) on \((x,y,t)\in\partial\omega\times(0,T]\) and initial conditions \(u(x,y,0)=\psi(x,y)\) on \((x,y)\in\overline\omega\), where \(\omega\) is a connected bounded domain in \({\mathbb R}^2\) with boundary \(\partial\omega\). The operator \(L\) is given by \(Lu=\varepsilon(u_{xx}+u_{yy})+b_1(x,y,t)u_x+b_2(x,y,t)u_y\) where \(\varepsilon\) is a small positive parameter and \(b_1,b_2,f,g,\psi\) are smooth functions. The monotone scheme leads to the existence and uniqueness of the solution and is shown to be uniformly convergent. The scheme is applied to reaction-diffusion and convection-reaction-diffusion problems and numerical comparison is made with other schemes.