Preconditioning of nonconforming finite element methods for second-order elliptic boundary value problems
This thesis deals with the condition numbers and singular value distribution of the preconditioned operators B[sub h][sup [minus]1]A[sub h] and A[sub h]B[sub h][sup [minus]1], where A[sub h] and B[sub h] are nonconforming finite element discretizations of second-order elliptic operators A and B. It generalizes the works of Manteuffel and Parter, Goldstein, Manteuffel and Parter, as well as Parter and Wong. Three nonconforming finite element methods are considered. The first two methods, the penalty method and the method of nearly-zero boundary conditions, deal with Dirichlet boundary conditions on curved domains. It is shown that if the leading part of A is a smooth function times the leading part of B and if the boundary condition of B is the same as that of A, then the L[sup 2]-condition number of A[sub h] B[sub h][sup [minus]1] is bounded independent of the mesh size h, and its L[sup 2]-singular values cluster and fill-in some bounded, estimable interval. If the boundary condition of the adjoint of B is the same as that of the adjoint of A, then the above conclusion holds for B[sub h][sup [minus]1]A[sub h] instead of A[sub h]B[sub h][sup [minus]1]. If B is self-adjoint and positive definite, then the B[sub h]-condition number of B[sub h][sup [minus]1]A[sub h] is bounded independent of h and the B[sub h]-singular values of B[sub h][sup [minus]1]A[sub h] cluster in some bounded, estimable interval. None of the results require full H[sup 2]-regularity nor uniform grids. The third method uses the P[sub 1] nonconforming finite elements, which consist of piecewise linear functions continuous only at mid-points of interelement boundaries. The generalized results of Parter and Wong are valid for this method with quasi-uniform grids and simpler boundary conditions. Through these three particular examples, one has learned what difficulties to anticipate when dealing with nonconforming finite element methods in general.
- Research Organization:
- Wisconsin Univ., Madison, WI (United States)
- OSTI ID:
- 5171555
- Resource Relation:
- Other Information: Thesis (Ph.D.)
- Country of Publication:
- United States
- Language:
- English
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