Resolution of Steep Internal Layers in One-Dimensional Problems

Accurately estimating solutions for models of physical systems with steep internal layers is an important but difficult numerical problem. For example, the formation of alloys—such as the creation of stainless steel—involves phase separation at the atomic level of its metallic components. This can be successfully modeled, but the delicate sharp interface structure makes computational methods for approximating solutions an ambitious problem. Sharp interface systems arise in the form of elliptic partial differential equations in an array of fluids problems, in semiconductor device simulations, and in microstructure evolution of materials.

The Research Spotlights paper in this issue, “Norm-Preserving Discretization of Integral Equations for Elliptic PDEs with Internal Layers I: The One-Dimensional Case,” by Travis Askham and Leslie Greengard, concentrates on the numerical problem of accurate resolution of steep internal layers in one-dimensional problems. Common approaches include adaptive finite difference, finite element, and volume integral methods. This paper concentrates on the third approach, and in particular on adaptive volume integral methods that preserve L p norm. It considers the effect of the choice of p on conditioning, showing in an example that the conditioning varies significantly with p. Well-conditioned discretization methods are shown to require both adaptive refinement within the steep internal layer and a careful choice of the value of p.

This paper will appeal to SIAM readers with an interest in numerical methods for partial differential equations, materials science, fluid dynamics, or semiconductor physics.

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Norm-Preserving Discretization of Integral Equations for Elliptic PDEs with Internal Layers I: The One-Dimensional Case

SIAM Review, 56(4), 623-623.