## 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.

### MSC:

 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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