Afrl-afosr-va-tr-2016-0015 Fast Implicit Methods for Elliptic Moving Interface Problems

tex Sat Dec 05 13:37:58 2015 1 Two notable advances in numerical methods were supported by this grant. First, a fast algorithm was derived, analyzed, and tested for the Fourier transform of pi ecewise polynomials given on d-dimensional simplices in D-dimensional Euclidean space. T hese transforms play a key role in computational problems ranging from medical imaging to partial differential equations, and existing algorithms are inaccurate and/or prohibitiv ely slow for d > 0. The algorithm employs low-rank approximation by Taylor series organi zed in a butterfly scheme, with moments evaluated by a new dimensional recurrence and sim plex quadrature rules. For moderate accuracy and problem size it runs orders of magnitud e faster than direct evaluation, and one to three orders of magnitude slower than the cla ssical uniform Fast Fourier Transform. Second, bilinear quadratures ---which numerically evaluate continuous bilinear maps, such as the L2 inner product, on continuous f and g belonging to known finite-dimensional fun ction spaces---were analyzed and developed. Such maps arise in Galerkin methods for diff erential and integral equations. Bilinear quadratures were constructed over arbitrary Ddimensional domains. In one dimension, integration rules of this type include Gaussian q uadrature for polynomials and the trapezoidal rule for trigonometric polynomials. A numer ical procedure for constructing bilinear quadratures was developed and validated. DISTRIBUTION A: Distribution approved for public release. FAST IMPLICIT METHODS FOR ELLIPTIC MOVING INTERFACE PROBLEMS