Original scientific paper
Basis of splines associated with singularly perturbed advection -diffusion problems
Tina Bosner
; Department of Mathematics, University of Zagreb, Zagreb, Croatia
Abstract
Among fitted-operator methods for solving one-dimensional singular perturbation problems one of the most accurate
is the collocation by linear combinations of $\{1,x,\exp{(\pm p x)} \}$, known as tension spline collocation. There exist
well established results for determining the `tension parameter' $p$, as well as special collocation points,
that provide higher order local and global convergence rates. However, if the advection-diffusion reaction problem
is specified in such a way that two boundary internal layers exist, the method is incapable of capturing
only one boundary layer, which happens when no reaction term is present. For a pure advection-diffusion problem
we therefore modify the basis accordingly, including only one exponential, i.e. project the solution to
the space locally spanned by $\{1,x,x^2,\exp{(p x)}\}$ where $p>0$ is the tension parameter. The aim of the paper
is to show that in this situation it is still possible to construct a basis of $C^1$-locally supported functions
by a simple knot insertion technique, commonly used in computer aided geometric design. We end by showing that
special collocation points can be found, which yield better local and global convergence rates, similar to the tension spline case.
Keywords
singular perturbations; advection-diffusion; Chebyshev theory; exponential tension splines; knot insertion
Hrčak ID:
53144
URI
Publication date:
10.6.2010.
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