Stabilities and error estimates for finite element approximations

Abstract

Finite element method is one of the main tools for the numerical treatment of elliptic partial differential equations. In this thesis, the stabilities and error estimates for finite element approximations is studied for the second order boundary value problem and the Stokes problem. The stability of the finite element method for the Stokes problem depends on the choice of finite element spaces for the velocity and the pressure. The finite element approximation scheme with divergence augmentation shows that the $P_{k+1}-P_{k-1}$ triangular elements, the $Q_{k+1}-Q_{k-1}$ quadrilateral elements in $\Real^2$, k ≥ 1, and the cross-grid $P_{k+1}-P_{k-1}$ tetrahedral elements in $\Real^3$, k ≥ 2, are stable. Also, the modified cross-grid element using continuous piecewise linear polynomials to approximate velocities and piecewise constants to approximate pressures is proved to be stable using the macroelement technique arguments. The mortar method as a new approach to domain decomposition which allows the coupling of nonmatching triangulations along interior interfaces between subdomains or discretization schemes is considered for second order elliptic problems. The nonconforming finite element on rectangular meshes with the local basis $\mbox{Span}\big\{1, x, y, \big(x^2-\frac{5}{3}x^4\big) -\big(y^2-\frac{5}{3}y^4\big)\big\}$ is used in each subdomain and the convergence of optimal order in the the broken energy norm is derived. Finally, numerical experiments which confirm the stabilities and the error estimates of optimal order are provided.