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Sparse spectral and p ‐finite element methods for partial differential equations on disk slices and trapeziums
Author(s) -
Snowball Ben,
Olver Sheehan
Publication year - 2020
Publication title -
studies in applied mathematics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.164
H-Index - 46
eISSN - 1467-9590
pISSN - 0022-2526
DOI - 10.1111/sapm.12303
Subject(s) - mathematics , partial differential equation , mathematical analysis , orthogonal polynomials , dirichlet boundary condition , boundary value problem , constant coefficients , orthogonal collocation , spectral method , biharmonic equation , ordinary differential equation , differential equation , collocation method
Sparse spectral methods for solving partial differential equations have been derived in recent years using hierarchies of classical orthogonal polynomials on intervals, disks, and triangles. In this work, we extend this methodology to a hierarchy of nonclassical orthogonal polynomials on disk slices and trapeziums. This builds on the observation that sparsity is guaranteed due to the boundary being defined by an algebraic curve, and that the entries of partial differential operators can be determined using formulae in terms of (nonclassical) univariate orthogonal polynomials. We apply the framework to solving the Poisson, variable coefficient Helmholtz, and biharmonic equations. In this paper, we focus on constant Dirichlet boundary conditions, as well as zero Dirichlet and Neumann boundary conditions, with other types of boundary conditions requiring future work.