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The DG methods are ideally suited for numerically solving hyperbolic problems. However this is not the case for diffusion problems,even though they are ideally suited for hp-adaptivity. Indeed, when compared with the classical continuous Galerkin methods on the same mesh, they have many more global degrees of freedom and they are not easy to implement. When compared with the mixed methods, they do not provide optimally convergent approximations to the flux and do not display superconvergence properties of the scalar variable. As a response to these disadvantages, the HDG methods were introduced in [6]. Therein, it was shown that HDG methods can be implemented as efficiently as the mixed methods. Later in [7] it was proven that the HDG methods do share with mixed methods their superior convergence properties while retaining the advantages typical of the DG methods. Inspired by these results, in this Thesis we are trying to explore HDG methods in a wider circumstance.

With the upturn in living standards and increasing work stress, people have started to pay more attention to how happy they feel in their daily lives. Evaluating people's happiness is becoming increasingly important in scientific fields. The focus of this thesis is to build a rational happiness function and to do simulations on several kinds of social networks related to the real world in a trading model with only two types of goods but no common currency. The purpose of the simulations is to seek the equilibrium points of the trading system. For this model, the parameters in the function and the shape of networks are key factors that affect the experimental results.