Very fine discretizations of differential operators often lead to large, sparse matrices A, where the condition number of A is large. Such ill-conditioning has well known effects on both solving linear systems and eigenvalue computations, and, in general, computing solutions with relative accuracy independent of the condition number is highly desirable. This dissertation is divided into two parts. In the first part, we discuss a method of preconditioning, developed by Ye, which allows solutions of Ax=b to be computed accurately. This, in turn, allows for accurate eigenvalue computations. We then use this method to develop discretizations that yield accurate computations of the smallest eigenvalue of the biharmonic operator across several domains. Numerical results from the various schemes are provided to demonstrate the performance of the methods. In the second part we address the role of the condition number of A in the context of multigrid algorithms. Under various assumptions, we use rigorous Fourier analysis on 2- and 3-grid iteration operators to analyze round off errors in floating point arithmetic. For better understanding of general results, we provide detailed bounds for a particular algorithm applied to the 1-dimensional Poisson equation. Numerical results are provided and compared with those obtained by the schemes discussed in part 1.