Algorithms for the accurate and efficient solution of fourth order boundary-layer problems
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2022-06-10Author
Alssaedi, Faiza
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Abstract
In this thesis, we study the analysis and numerical solution of second-order complex-valued reactiondiffusion equations, and two families of fourth-order singularly perturbed problems. The problems
are all singularly perturbed, meaning that each has a parameter, ε, multiplying the highest derivative. This parameter is positive but maybe arbitrarily small. However, as ε Ñ 0, the differential
equations become ill-posed, hence the singular nature of the perturbation.
The first problem we address is the numerical solution, by finite difference methods of a secondorder, complex-valued problem. We employ specialised fitted meshes: the well-known piecewise
uniform Shishkin mesh, and the graded Bakhvalov mesh. The numerical analysis of such methods
usually rely on maximum principles, but these do not hold, in a direct way, for complex-valued
problems. So we present an approach for rewriting the equation as a coupled system of real-valued
problems, and establish that the coefficient matrix for this system is positive definite. Then we
show how to adapt the analysis of Bakhvalov [2], in the style of Kellogg et al. [20], to prove
convergence.
The second problem we address is the numerical solution of a fourth-order, real-valued reactiondiffusion problem. The ODE is “simply supported” (see Section 1.5.3), and so has boundary
conditions that allow it to be transformed into a (weakly) coupled system of second-order reactiondiffusion equations, involving unknowns related to the solution to the fourth-order problem, and
its second derivative.
When analysing a finite element method for solving this system, it is usually assumed that the
coupling matrix of the coupled system is pointwise coercive. However, we show that the standard
transformation (see, e.g., [47]) cannot satisfy this condition. This motivates us to propose a new
transformation which resolves this issue.
Moving on to finite difference methods for this problem, we show how to adapt the transformation in a way that leads to a maximum principle-type result. Moreover, we present an iterative
scheme for solving the continuous problem in order to derive a stability result for the differential
operator. The convergence of the finite difference scheme on a Shishkin mesh, then follows from
standard arguments.
Finally, we address the numerical solution using a fourth-order, complex-valued reactiondiffusion problem. We extend the transformation from earlier sections to deal with this case,
again focusing on how to ensure coercivity (for a finite element method) and monotonicity (for
analysis of a finite difference scheme).
Through all these sections, numerical results are presented that verify the convergence of the
schemes, and test if the theoretical orders of convergence are observed in practice.