On Extension of Boundary Algebraic Equations to Irregular Lattices

This study is concerned with extending the use of boundary algebraic equations (BAEs) to problems involving irregular rather than regular lattices. Such an extension would be indispensable for solving multiscale problems defined on irregular lattices, as BAEs provide seamless bridging between discrete and continuum models. BAEs share many features with boundary integral equations and are particularly effective for solving problems involving infinite domains. However, it is shown that, BAEs for irregular lattices containing certain terms may require the same amount of computational effort as the original problem for which the equations are formulated. In this paper, we formulate a BAE for a model problem and expose the fundamental obstacle that prevents us from using that BAE as an effective tool. It is shown that, in contrast to regular lattices, BAEs for irregular lattices require a statistical rather than deterministic treatment. This is a very interesting direction for future research.

Boundary algebraic equations for lattice problems

Procedures for constructing boundary integral equations equivalent to linear boundary-value problems governed by partial differential equations are well established. In this paper, it is demonstrated how these procedures can be extended to linear boundary-value problems defined on lattices and governed by algebraic (‘difference’) equations. The boundary equations that arise are then themselves algebraic equations. Such ‘boundary algebraic equations’ (BAEs) are derived for fundamental boundary-value problems defined on both perfect lattices and lattices with defects. It is demonstrated that key advantages of representing a continuum boundary-value problem as an equation on the boundary, such as favourable spectral properties and minimal problem size, are preserved in the lattice environment. Certain spectral properties of BAEs are established rigorously, whereas others are supported by numerical experiments.