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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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