Convection Diffusion equation

Before moving to more complex numerical examples, preliminary tests are done on a case with a known closed form solution. As in all the previous examples, we use the diffusion equation for a test case. The main purpose of this section is basic evaluation of mesh based against meshless methods, as well as strong form against weak form methods. More specific analyses and performance tests are presented in Chapter \ref{cases}.

\section{Diffusion equation}

\index{Diffusion equation} A 2D diffusion equation in its dimensionless form is considered: \begin{align} %??? brezdimenzijska oblika nima c u_{,t} - \nabla^2u = 0, & \qquad (x, y)\in \Omega, \label{eq.diffu}\\ % \frac{\partial u(x,y,t)}{\partial t} = \nabla^2 u(x,y,t), & \qquad (x, y)\in \Omega, \label{eq.diffusion}\\ u(x,y,t) = \overline{u}(x,y,t), & \qquad (x,y) \in \Gamma_D,\label{eq.bc_diffu}\\ u(x,y,t),_n = \overline{g}(x,y,t), & \qquad (x,y) \in \Gamma_N, \label{eq.mixed.bc_diffu}\\ u(x,y,t) = u_0, & \qquad t=0,\label{eq.ic_diffu} \end{align} where $(x,y)$ are spatial coordinates, $t$ is time, $u(x, y, t)$ is the unknown solution, $ \Omega$, and $ \Gamma_D$ and $ \Gamma_N$ are the global domain with Dirichlet boundary and Neumann boundary, $\overline{u}$ and $\overline{g}$ are the prescribed Dirichlet and Neumann boundary values and $u(x,y,0)=u_0$ is the known initial condition.