James Adler
A Non-symmetric Least-Squares Finite-Element Method for Singularly Perturbed Reaction-Diffusion

503 Boston Ave
Medford
MA 02155
james.adler@tufts.edu
Scott MacLachlan
Niall Madden

In this talk, we consider the numerical solution of a singularly perturbed reaction-diffusion problem:

$$Lu:=-\varepsilon ^2 \Delta u + b u = f\quad{\rm in\ }\Omega: = (0,1)^2, \qquad u = 0 \ \partial \Omega.$$

where $\varepsilon$ is a small positive parameter. The solution exhibits boundary layers with width of order $\varepsilon$ . Our goal is to generate a numerical solution using a finite-element method on a tensor-product mesh, which accurately solves the problem and resolves the layers, but without any assumption that the number of nodes in each dimension, $N$ , is ${\mathcal{O}}(\varepsilon ^{-1})$ .

Using a standard Galerkin method, it is natural to conduct the analysis with respect to the $\varepsilon$ -weighted norm (energy norm), $\Vert v \Vert _\varepsilon = \big( \varepsilon ^2 \vert v\vert^2_1 + \Vert v\Vert^2_{0}\big)^{1/2}$ . This has been done in several studies, such as [2], where the method is implemented with bilinear elements on a specially constructed piecewise-uniform Shishkin mesh. However, the energy norm is weak for this problem, since $u_x$ and $u_y$ are both ${\mathcal{O}}(\varepsilon ^{-1})$ . Thus, as $\varepsilon \to 0$ , $\Vert u \Vert _\varepsilon \to \Vert u\Vert _0$ . This is particularly problematic, since it is easy to show that, for sufficiently small $\varepsilon$ , if the FEM solution, $u^N$ , is computed on a uniform mesh, then $\Vert u-u^N\Vert _\infty$ is ${\mathcal{O}}(1)$ (meaning there is no convergence, since the layers are not resolved), yet $\Vert u-u^N\Vert _0 \sim N^{-1/2}$ . Even with the layer resolving Shishkin mesh, the error estimate in the energy norm is $\varepsilon$ -dependent: $\Vert u-u^N \Vert _\varepsilon \sim N^{-2} + \varepsilon ^{1/2} N^{-1} \ln N$ .

More recently, Lin and Stynes [1] have proposed the use of a ``balanced norm'', which arises in the analysis of a mixed finite-element method and requires $H(\operatorname{div})$ conforming spaces. In this talk we extend these results in the framework of a non-symmetric least-squares formulation, which finds the solution in an $H^1$ conforming finite-element space. The resulting finite-element solutions yield $\varepsilon$ -independent convergence in a balanced norm, resolving the boundary layers accurately and efficiently, yet are also amenable to solution using scalable algebraic multigrid methods.