An Adaptive Finite Element Algorithm For The Heat Conduction Equation

This paper has designed an adaptive finite element algorithm for the heat conduction equation. To control the posteriori error estimator in the temporal direction, we applied the energy norm of the finite element solutions of two neighbored time steps and the oscillation of the right hand f(x). In the spacial direction, we applied the SPR posteriori error estimator which is widely used for the steady-state problems. In the mesh refinement aspect, we used the newest rule of vertex bisection, which can make sure that the meshes are regular. Finally, the numerical examples showed that the choice of our estimators in the two directions is correct and the whole algorithm is valid. Therefore our adaptive algorithm is successful.The innovation of this paper is that we have used the general SPR estimator in heat conduction problems. We have realized that the temporal step and the spacial mesh are refined and coarsen automatically, so we obtain the adaptive finite element algorithm for the parabolic question. What's more, our numerical examples offer a strong evidence to show that our algorithm is very effective.