Numerical Methods For Three Kinds Of Boundary Value Problems For Differential Equations

Numerical methods for three kinds of boundary value problems for differential equations are discussed. A stabilized mixed finite element method is proposed and analyzed for the generalized Stokes equation by employing the mini finite element. Stability is obtained when the velocity space is equipped with a new defined norm. A practicing scheme with numerical integration is presented since the integrals are seldom computed precisely. A numerical method for solving the circular arch problem is considered. To get an approximate solution of this problem, a smoothing finite element method is given with cubic spline spaces. Error estimation can reach a higher convergence order. Based on the boundary condition of the fourth-order problem, a Fourier spectral method for solving the biharmonic boundary problem is proposed and analyzed by using trigonometric function and bubble function. The numerical solution is smooth enough and the convergence is proved.