In this paper, we consider the Characteristics-Finite Volume Element Methodfor the convection dominated diffusion equation and Expanded Mixed Finite ElementMethod for the initial-value problems of purely longtudinal motion of a humogeneousbar, and obtained the error estimates of this two discrete solutions.In chapter one, we consider the Characteristics-Finite Volume Element Methodfor the following convection dominated diffusion equationTheoretical analysis indicates that the method is effective and has optimal error esti-mates in L~2 and H~1 norm.In chapter two, we consider the Expanded Mixed Finite Element Method for thefollowing initial-value problems of purely longtudinal motion of a humogeneous barWe present the error analysis of the solutions of the expanded mixed finite elementscheme and derive optimal error estimates for the discrete scheme in L~2-norm and inH~1-norm.
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