Discretization Of Elliptic Problems And Its Effective Solver

The object of this thesis is to discuss the construction and the convergence analysisof the discretization of elliptic problems and its effective solvers, which includes thefollowing four aspects.Firstly, a new finite volume method based on P₁ nonconforming quadrilateral fi-nite element for second order elliptic boundary value problems is presented and ana-lyzed. The optimal broken H¹ and L² error estimates are proved without the conditionon the partition which requires that the distant d_K between the midpoints of the twodiagonals is of order O(h²) for all quadrilateral element K as h goes to zero.Secondly, the a prior error estimate of cell boundary element method is improvedin the case of triangular partition, which is proved to be optimal in H¹ norm sense.Thirdly, the local and parallel two-grid algorithms for some nonconforming finiteelement and finite volume discretization are discussed. A kind of feasible way to con-struct the intergrid transfer operator is proposed. It is proven that the convergence ratesare optimal in broken H¹ norm sense for P₁ nonconforming triangular finite elementand P₁ nonconforming triangular finite volume methods. Furthermore, by combiningthis algorithm with partition-of-unity technique, an improved local and parallel two-grid algorithm is proposed. Convergence analysis and numerical experiments indicatethat the algorithm can successfully improve the accuracy by O(h^1/2) order.Finally, the cascadic multigrid methods for P₁ nonconforming quadrilateral finiteelement and P₁ nonconforming quadrilateral finite volume schemes are presented andanalyzed. A new grid refinement and a new grid transfer operator are constructed. It isshown that these algorithms possess the optimal accuracy and computational complex-ity with conjugate gradient, Jacobi and Gauss-Seidel smoothers.