| In the current process of scientific and engineering computing, finite element method because of its superiority played an extremely important role in solving partial differential equations. The adaptive finite element method is a high efficiency, high reliability calculation method, it is based on finite element method, its core is posteriori error estimation and adaptive mesh improvement techniques. The process can be described with the following cycle:Solving equations→posteriori error estimation→tag→mesh refinement,or coarseningThe essence of adaptive method is that takes estimation operator as the basis which is obtained by a posteriori error estimates, to achieve grid local refinement. Its main principle is refinement grid in the place of indicating a large estimation oper-ator, so grid point distribution in the region where the function regularity is poor is more dense. Based on this, whether the estimation operator is valid and reliable method for adaptive element method is an important criteria. So how to choose estimation operator for a posteriori error estimates is especially important.We consider that a class of strongly nonlinear coupled thermal problems:Firstly,the article shows some properties of the heat equations:existence of weak solutions, regularity and convergence. It also shows a priori error estimates of the problem on this model. Based on this,according to the theoretical framework of a posteriori error estimation of nonlinear problems, this nonlinear coupled thermal problems were given a optimal a posteriori error estimates in the W1,p×W1,q norm and Lp×Lq norm, which can be used to guide adaptive mesh encryption. This is the main part of this article.The innovation point of this article is that gives a posteriori error estimates in the W1,p×W1,q norm and Lp×Lq norm by choosing different finite element space and the norm. In the numerical experiments, we can adjust the grid on the basis of a posteriori error estimates.
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