Robust error estimates in a balanced norm for the approximation of reaction--diffusion equations on graded meshes

ARMENTANO, MARÍA G.; LOMBARDI, ARIEL L.; PENESSI, CECILIA

Buenos Aires

Conferencia; International Conference on Boundary and Interior Layers; 2022

For the reaction--diffusion problem\begin{eqnarray*} -\varepsilon^2\Delta u + b(x)u &=&f\qquad \mbox{in }\Omega\\ u&=&0\qquad\mbox{on }\partial\Omega\end{eqnarray*}where $b(x)\ge b_0>0$ on $\Omega$ and $\varepsilon$ is a small positive parameter, we consider the weighted variational formulation introduced in [N. Madden, M. Stynes, Calcolo, vol. 58, 2021]: find $u\in H^1_0(\Omega)$ such that\[ B_\beta(u,v) = \varepsilon^2\left(\nabla u,\nabla(\beta v)\right) + \left(b(x)u,\beta v\right) = \left(f(x),\beta v\right)\qquad\forall v\in H^1_0(\Omega)\]where the weight $\beta$ is defined by\[ \beta(x) = 1 +\frac1\varepsilon \exp\left(-\gamma\frac{d(x)}\varepsilon\right), %qquad x\in\Omega.\]with $d(x)$ the distance from $x$ to the boundary of $\Omega$ and $0