Overview

As a prelude to the error analysis of various numerical schemes for solving the primal variational problem, we will first give a convergence analysis and derive error estimates for numerical solutions of the abstract problem, introduced in Chapter 7, which includes the primal variational problem as a special case. In the next chapter, we will apply the results presented here to perform an error analysis for various numerical approximation schemes for solving the primal problem. For convenience, let us recall the abstract problem.

Problem Abs. Find w: [0, T] ? H, w(0) = 0, such that for almost all t ? (0, T), and

Under the assumptions that

• H is a Hilbert space

• K ? H is a nonempty, closed, convex cone

• a: H x H ? is a bilinear form on H, symmetric, bounded and H-elliptic

• ? ? H ¹ (0, T; H'), ?(0) = 0

• j: K ? is nonnegative, convex, positively homogeneous, and Lip-schitz continuous

we have the existence of a unique solution w ? H ¹(0, T; H) of the problem Abs. For convenience, later on we will refer to these assumptions as the standard assumptions for the problem Abs. In this chapter we will always assume that these standard assumptions hold. We also recall that there exists w* ? H ¹((), T; H') such that

In the first three sections...