Abstract
We consider a singularly perturbed reaction diffusion problem as a first order two-by-two system. Using piecewise discontinuous polynomials for the first component and H_{{{,mathrm{{div}},}}}-conforming elements for the second component we provide a convergence analysis on layer adapted meshes and an optimal convergence order in a balanced norm that is comparable with a balanced H^2-norm for the second order formulation.
Highlights
Consider the singularly perturbed reaction diffusion problem, given in Ω = (0, 1)2 by− ε2Δu + cu = f, (1)where 0 < ε 1, c ∈ W 1,∞, c∞ ≥ c ≥ c0 > 0 and u = 0 on ∂Ω
Over the last years convergence in a balanced norm, where the boundary layers do not vanish for ε → 0, was considered, see [1,9,11,17]
Remark 2 Using above solution decomposition for u we derive a similar decomposition for the solution U of (3), as U = (u, u) and u = −ε grad u
Summary
Consider the singularly perturbed reaction diffusion problem, given in Ω = (0, 1) by. We rewrite the problem, using u = −ε grad◦ u, into a first order system c0 01. Where grad◦ denotes the gradient in H01(Ω) and div its adjoint, the divergence. This formulation is called a mixed formulation. For its weak formulation let ·, · denote the L2-scalar product over Ω. (2) becomes with V = (v, v) ∈ L2(Ω) × Hdiv(Ω)
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