M.Sc Student | Sherman Alexander |
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Subject | High-Order Spectral Methods for Elliptic PDE |

Department | Department of Applied Mathematics |

Supervisor | Professor Nadav Liron |

In this thesis we deal with the solution of Elliptic PDE in rectangular domains in two dimensions by pseudo-spectral methods. First we treat constant coefficient Poisson and Helmholtz equations with Diriclet boundary conditions. The treatment of these types of equations is based on the relation between the Fourier coefficients of the solution and those of the right hand side of the equation (RHS). An important feature of the present method is that it treats explicitly the singular points in order to reserve a high order of accuracy. Typically, solution to elliptic problems in rectangular regions have singularities at the corner points which may appear even if the source function is smooth inside the region and the boundary functions are continuously differentiable on each side of the rectangle. The presence of singularities in the solution significantly degrades the quality of any numerical representation, especially for high order methods. To address such discontinuities we deploy a subtraction technique, based on presenting the RHS as sum of a smooth periodic function and an auxiliary function which can be integrated analytically. We generalize this technique to higher orders of accuracy, resulting in a highly accurate Fourier-spectral solver. It is to be emphasized that the subtraction methods can be incorporated with any numerical scheme, when the

boundary conditions degrade the accuracy, as far as appropriate subtraction functions can be found. From this point of view, the present work paper is an illustration of how the subtraction technique could be successfully incorporated in the Fourier 2-D algorithm for the solution of the Poisson and modified Helholtz equations.

In addition, we develop a solver for non-separable, self adjoint elliptic equations with variable coefficients. Two non-constant cases were considered and special transformations for each case were introduced. The special transformations applied to the variable coefficients equations enable a reduction of the non-constant equation to an auxiliary constant coefficient Poisson or Helmholtz equation with only a few correction steps. Numerical experiments with simulation software, developed throughout the thesis, led to application of the domain decomposition (DD) technique in order to achieve a low number of correction steps and to improve the accuracy in the whole domain. The efficiency of DD using the transformation to the constant coefficient case is demonstrated in 1-D case only due to matching problems in the 2-D case.