Remarks on balanced norm error estimates for systems of reaction-diffusion equations. (English) Zbl 1499.65353

The numerical solution of systems of reaction-diffusion equations \(-Eu^{\prime\prime}+Au=f\) in \(\Omega =(0,1)\) and \(u=0\) on \(\partial\Omega\) is considered. Hereby, \(E\) is a diagonal matrix, where the diagonal elements \(\varepsilon_i\), \(i=1,2,\ldots,\ell\), are small real parameters, and \(A\) is a symmetric, strictly diagonally dominant matrix with positive entries on the main diagonal. Furthermore, the entries of the matrix \(A\) and the function \(f\) are sufficiently smooth. This problem is discretized by means of the finite element method with linear elements on a Shishkin mesh. It is discussed how one can prove error estimates in the balanced norm \(\| v\|_b^2=\sum_i\varepsilon_i^{1/2}(v_i^\prime,v_i^\prime)+\| v\|_0^2\), where \((\cdot,\cdot)\) and \(\| \cdot \|_0\) are the \(L_2\)-inner product and the \(L_2\)-norm, respectively. The basic ideas for getting such estimates are discussed in two cases. For simplicity systems of two equations are considered. At first the case \(\varepsilon_1=\varepsilon_2\) is discussed. Then, the case with \(\varepsilon_1\ne\varepsilon_2\) and constant coefficients is considered.


65L60 Finite element, Rayleigh-Ritz, Galerkin and collocation methods for ordinary differential equations
65L70 Error bounds for numerical methods for ordinary differential equations
34D15 Singular perturbations of ordinary differential equations
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