Stability analysis of convection–diffusion equations of different finite-element spaces at discrete times

Abstract

The convection-controlled diffusion problem is hyperbolic in nature and its solutions tend to have numerical shocks. To solve the problem of time dependence, it is common to use finite-element method in the spatial region, while the algorithm proposed in the past with finite differential procedure is mostly limited to the fixed finite-element grid of the spatial region. We often need to use different finite-element spaces at different times, such as the spread of flame, oil and water frontier problems; so many mathematicians and engineers have set their sights on the use of dynamic finite-element space, but also put across a lot of dynamic finite-element methods in giving the general parabola problem of the variable-mesh finite-element method. The principal purpose of this paper is to adopt different spatial grids for different time layers, and project the approximate solution of the previous time layer to the present time layer to act as the initial value of the current layer to enable us deduce the stability at discrete times.