The Solution of Partial Differential Equations on Realistic Geometries

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Many useful physical quantities (capacitance, stresses, electromagnetic scattering, etc.) are computed by solving elliptic partial differential equations (PDEs) on realistic geometries with boundaries containing corners, edges, and conical points. Even for simple boundaries (i.e. a cube), the solutions to the PDEs on the corresponding regions usually have singularities near such features. This non-smooth behavior is a major sticking point in both the numerical solution of elliptic PDEs, and their mathematical theory. Numerically, non-smooth behavior can present the following problem. While smooth functions are usually representable to high precision by short finite Fourier series (or short series of Chebyshev or Legendre polynomials), singular functions can take on a bewildering variety of behaviors. For example, an analytic function can have poles, branches, or essential singularities (in the neighborhood of which it takes on every complex value except possibly one!). Often such functions can be represented by nested Chebyshev or Gauss-Legendre discretizations, but a large number of degrees of freedom is usually required to capture all of the possible unknown behavior to high precision. It turns out, however, that many of the singular functions encountered on geometries with corners and edges can be characterized in great detail. For instance, in the case of Laplace's equation on a two-dimensional domain with corners, the singular solutions near corners are representable by elementary asymptotic series of known singular powers. With such a representation in hand, the behavior of the solutions to the PDEs becomes significantly more circumscribed, and so efficient special-purpose discretizations can be constructed for the singular solutions. As a result, many historically numerically refractory PDEs involving domains with corners can be solved rapidly and to essentially machine precision. When solving elliptic PDEs numerically to high precision, it is often necessary to reformulate the problems as second kind integral equations using classical potential theory. In two dimensions, the solutions to the associated integral equations near corners have been characterized for several elliptic PDEs, however the much more detailed (and useful) case of three dimensions remains largely unexplored. We propose to construct an analytical apparatus characterizing precisely the behavior of the solutions to the integral equations associated with various elliptic PDEs (Laplace, Helmholtz, Stokes, and eventually Maxwell) in the vicinity of edges, corners, and conical points in three dimensions. We will construct a numerical apparatus exploiting this analytical information, obviating the need for nested discretizations. The creation of such boundary integral based schemes in three dimensions, if eliminating the longstanding (and numerically intractable) issues surrounding edges, would constitute a major advance in engineering and applied sciences.