#### Title

Sparse Spectral-Tau Method for the Two-Dimensional Helmholtz Problem Posed on a Rectangular Domain

#### Publication Date

Summer 6-2-2022

#### Abstract

Within recent decades, spectral methods have become an important technique in numerical computing for solving partial differential equations. This is due to their superior accuracy when compared to finite difference and finite element methods. For such spectral approximations, the convergence rate is solely dependent on the smoothness of the solution yielding the potential to achieve spectral accuracy. We present an iterative approach for solving the two-dimensional Helmholtz problem posed on a rectangular domain subject to Dirichlet boundary conditions that is well-conditioned, low in memory, and of sub-quadratic complexity. The proposed approach spectrally approximates the partial differential equation by means of modal Chebyshev integration matrices. Implementation of the boundary conditions is achieved through a technique known as ``integration preconditioning," although we refer to the technique as integration sparsification. The spectral method presented represents certain partial differential operators in terms of sparse, banded integration matrices. In this work, there are $N+1$ Chebyshev modes associated with each coordinate direction. Therefore, there are $n = (N + 1)^2$ modes in total. For the truncations considered, our method empirically yields a linear set-up cost, followed by a sub-quadratic solve complexity of $\mathcal{O}(n^{1.6})$.

#### Degree Name

Mathematics

#### Level of Degree

Masters

#### Department Name

Mathematics & Statistics

#### First Committee Member (Chair)

Stephen Lau

#### Second Committee Member

Deborah Sulsky

#### Third Committee Member

Jacob Bayer Schroder

#### Language

English

#### Keywords

Spectral-Tau Method, Helmholtz Equation, Numerical Methods, Partial Differential Equations, Integration Sparsification, Preconditioning, Iterative Methods, Chebyshev Polynomials

#### Document Type

Thesis

#### Recommended Citation

Dalton, Gabriella M.. "Sparse Spectral-Tau Method for the Two-Dimensional Helmholtz Problem Posed on a Rectangular Domain." (2022). https://digitalrepository.unm.edu/math_etds/171

#### Included in

Mathematics Commons, Partial Differential Equations Commons, Statistics and Probability Commons