| We develop a model for mass transport phenomena in microelectronic interconnect lines, study its mathematical properties, and present a number of new numerical methods which are useful for simulating the process.; The central focus of the work is an investigation of the well-posedness of the grain boundary diffusion problem. The equation is stiff and non-local and depends on gradients of the stress components σij, which contain singularities near corners and grain boundary junctions. We show how to recast the problem as an ODE on a Hilbert space, and prove that the non-selfadjoint compact operator involved has a real non-negative spectrum and a set of eigenfunctions which is dense in L2(Γ), where Γ is the grain boundary network. We show that in the limit of an infinite interconnect line, the eigenfunctions are sines and cosines, and through numerical studies show that for a finite line, in spite of non-orthogonality, the eigenfunctions form a well conditioned basis for L 2(Γ). We develop a numerical method for simulating the evolution process, and obtain results which are self-consistent under mesh-refinement and make physical sense.; The main tool we develop for solving the elasticity equations with high resolution near corners and grain boundary junctions is a singularity capturing extension of the First Order System Least Squares finite element method. We call the method XFOSLS. The method is designed for polygonal domains with corners, cracks, and interface junctions, and can handle complicated jump discontinuities along interfaces while maintaining smooth displacement and stress fields (which may be unbounded near corners) within each region of the domain. Several self-similar (not necessarily singular) solutions may be adjoined to each corner or junction, and the supports of the extra basis functions may overlap one another.; We also present a number of new techniques for computing bases of self-similar solutions for corners and interface junctions. These include an algorithm for removing rank deficiency from the basis matrix at isolated parameter values, a stabilization algorithm for removing near linear dependency from the power solutions corresponding to characteristic exponents which are clustered together, and a new theorem about Keldy˘s chains which leads to an algorithm for computing associated functions. Several examples are given which illustrate the techniques. |
