Convergence Analysis Of An Adaptive Finite Element Method For The Helmholtz Equation In L~2 Norm

This dissertation addresses the a posteriori error estimation of the finite element method and convergence analysis of the adaptive algorithms for the Helmholtz equation with high wave number in L2 norm. Different from previous related research work which analyzed the results in energy norm, our work is based on the L2 norm.In Chapter 2, we introduce the refinement procedure which guarantees the G-gradedness of the partition and the content of the "mesh functions" hT presented in [12]. In Chapter 3, we introduce the model problem of the Helmholtz equation and its finite element discretization, and we also derive the a posteriori error estimates in L2 norm of residual-type. In Chapter 4, we first introduce the adaptive finite elemen-t method(AFEM), and then establish the control relationship between energy norm |||hT(u-uT)||| and L2 norm error||u - rT||L2(Ω)·Finally, we prove the convergence of the AFEM in L2 norm.