Convergence Of Adaptive Finite Element Methods For Static Temperature Control Problem

In present paper, we focus on two problems. They will be discussed in two parts respectively.In the first part, we concentrate on the Convergence of Adaptive Fi-nite Element Methods for Static Temperature Control Problem. Adaptive Finite Element Methods (see [2][5][6]) for numerically solving optimal control problems are often used in practice. In this part, we concentrate on the Static Temperature Control Problem. At the beginning, we discuss a posteriori error estimates for the standard finite element approxima-tion of the model problem in L2-norm and H1-norm respectively. Then based on the above results, we obtain an error reduction rate. Further-more, together with a reduction rate of data oscillation, we construct an adaptive FEM algorithm for the model problem. Next, we prove that this algorithm is convergent.In the second part, we work on the Pressure diffusion behavior of the gas in nanomaterials. Gas can penetrate any solid matter more or less, and the diffusion rule of gas in solid is similar to its self-diffusion. Dif-fusion coefficient is the basic parameter, which is decided by the nature of the gas-solid combination. In different solid, the diffusion coefficient of gas is different as well. As an application, for an unknown kind of solid, to study the diffusion coefficient of gas in it can help us understand its structure. By simulating the pressure diffusion behavior of the gas in nanomaterials, we establish the corresponding partial differential equa-tions, and we try to calculate the diffusion coefficient. With knowledge of physics, the problem above can be transformed into optimization prob-lem. Because the data in this experiment is discrete, we have to solve the problem using numerical methods (see [29][30][31][32]). Optimization algorithm (see [26][28][33]) is needed also.