On The Application Of Negative Norm Estimate In The Optimal Error Analysis Of Finite Element Methods

In this thesis,we establish an optimal error estimates for the Lagrange finite element methods(FEM)for an advection-diffusion-reaction equation system.In this advection-diffusionreaction model,the three chemical species are coupled through a first-order reaction term.Specifically,a product C is generated by two reactants A and B.In many applications,the density evolution of C is more important,which requires higher resolution.Therefore,in our numerical method the conventional r-th Lagrange finite element space Pr is applied to solve the equations of A and B,while Pr+1 is used to solve the equation of C to achieve more accurate numerical solutions.For the combination of(Pr,Pr,Pr+1)with r>1,we prove that(r+2)-th optimal L2-norm error estimate holds for the equation of C.That is,using one order lower finite element spaces for A and B will not influence the optimal(r+2)-th order convergence for C.Our proof is based on the negative norm estimates for Lagrange FEMs.Furthermore,we propose a simple one-step recovery technique to obtain a new numerical solution for A and B of(r+2)-th order accuracy at any given time step.Compared with using Pr+1 for all three unknowns,the proposed method with recovery enjoys a much smaller cost while also obtain(r+2)-th order accurate numerical solution for all variables.Numerical experiments in both two and three-dimensional spaces are provided to confirm our theoretical results and demonstrate the efficiency of the one-step recovery technique.