## Uniform approximation of singularly perturbed reaction-diffusion problems by the finite element method on a Shishkin mesh.(English)Zbl 1026.65106

The authors consider the two-dimensional singularly perturbed problem $-\varepsilon^2\Delta u+ u= f\quad\text{in }\Omega\subset \mathbb{R}^2,\quad u=0\quad\text{on }\partial\Omega,$ where the parameter $$\varepsilon$$ can approach zero, $$\partial\Omega$$ is smooth and $$f\in H^{4M+2}(\Omega)$$ for some $$M\in \{0,1,2,\dots\}$$. Let $$u_\varepsilon\in H^1_0(\Omega)$$ denote the unique solution to the following variational formulation of the above problem $B(u_\varepsilon, v)= F(v),\;\forall v\in H^1_0(\Omega),$ where $B(u,v):=\int_\Omega(\varepsilon^2\nabla u\cdot\nabla v+ uv) dx dy,\;F(v):= \int_\Omega fvdx dy$ and let $\Omega_0:= \{z- \rho\overline n_z; z\in\partial\Omega, 0<\rho\rho_0\},$ where $$\rho_0$$ is a positive constant less than the minimum radius of curvature of $$\partial\Omega$$ and $$\overline n_z$$ is the outward unit normal at a point $$z\in\partial\Omega$$.
Introducing the so-called Shishkin mesh $$\Delta_0$$ in $$\Omega_0$$ and a quasi-uniform mesh $$\Delta_1$$ in $$\Omega_1:= \Omega\setminus\Omega_0$$ compatible with $$\Delta_0$$ the authors obtain the following approximation of $$u_\varepsilon$$ by its finite element counterpart $$v_N\in S_N$$, uniformly with respect to $$\varepsilon$$, namely $\|u_\varepsilon- v_N\|_{\varepsilon,\Omega}\leq CN^{-p/2}[\ln(n+1)]^p,$ where $$C\in\mathbb{R}$$ does not depend of $$\varepsilon$$ and $$N$$. Here $$S_N$$ is the finite element space on $$\Delta_0\cup\Delta_1$$ of piecewise polynomials of degree $$\leq p$$, $$N= \dim S_N$$, $$n$$ is the number of subdivisions within $$\Omega_0$$ in the direction normal to the boundary and the energy norm $$\|\cdot\|_{\varepsilon,\Omega}$$ is defined by $\|\cdot\|^2_{\varepsilon,\Omega}:= \varepsilon^2|\cdot|^2_{1,\Omega}+\|\cdot \|^2_{0,\Omega},$ $$\|\cdot\|_{k,\Omega}$$ and $$|\cdot|_{k,\Omega}$$ being the norm and seminorm in $$H^k(\Omega)$$, respectively.
Numerical examples are presented giving high agreement of numerics with theory.
Reviewer: S.Burys (Kraków)

### MSC:

 65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs 65N15 Error bounds for boundary value problems involving PDEs 35J25 Boundary value problems for second-order elliptic equations 35B25 Singular perturbations in context of PDEs 65N50 Mesh generation, refinement, and adaptive methods for boundary value problems involving PDEs
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