## Dissertations, Master's Theses and Master's Reports

2019

#### Document Type

Open Access Dissertation

#### Degree Name

Doctor of Philosophy in Mathematical Sciences (PhD)

Department of Mathematical Sciences

Mark S. Gockenbach

Hassan Masoud

Benjamin W. Ong

Jiguang Sun

#### Abstract

Let $X$, $Y$, and $Z$ be real separable Hilbert spaces, let $T:X \to Y$ be a compact operator, and let $L:D(L) \to Z$ be a closed and densely defined linear operator. Then the generalized singular value expansion (GSVE) is an expansion that expresses $T$ and $L$ in terms of a common orthonormal basis. Under certain hypotheses on discretization, the GSVE of an approximate operator pair $(T_j,L_j)$, where $T_j:X_j \to Y_j$ and $L_j:X_j \to Z_j$, converges to the GSVE of $(T,L)$. Error estimates establish a rate of convergence that is consistent with numerical experiments in the case of discretization using piecewise linear finite elements. Further numerical testing suggests that a higher rate of convergence is attained by using higher order elements. However, the theory does not cover this case.

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