Robust error estimates in a balanced norm for the approximation of reaction-diffusion equations on graded meshes

GABRIELA ARMENTANO; ARIEL LUIS LOMBARDI; CECILIA PENESSI

Buenos Aires

Conferencia; International Conference on Boundary and Interior Layers; 2022

Universidad de Buenos Aires

For the reaction–diffusion problem−ε^2∆u + b(x)u = f in Ωu = 0 on ∂Ωwhere b(x) ≥ b0 > 0 on Ω and ε is a small positive parameter, we consider the weighted variational formulation introduced in [N. Madden, M. Stynes, Calcolo, vol. 58, 2021]: find u ∈ H^1_0(Ω) such thatB_β(u, v) = ε^2(∇u, ∇(βv)) + (b(x)u, βv) = (f(x), βv) ∀v ∈ H^1_0(Ω)where the weight β is defined by β(x) = 1 + 1/εexp(−γd(x)/ε),with d(x) the distance from x to the boundary of Ω and 0 < γ ≤ b_0.The bilinear form B_β is coercive and continuous in the weighted norm|||v|||_β =(ε^2||∇v||^2_β + ||v||^2_β)^(1/2)where ||v||_β = (βv, v)^(1/2). It turn out that this norm is balanced for the problem under consideration.In the case of Ω being a rectangle, we consider the approximation by continuous piecewise bilinear functions based on this variational formulation on graded meshes, which depend on a graduation parameter which controls how refined becomes the mesh near the boundary layers. We prove that, by using appropriate graded meshes, we obtain numerical solutions with error estimates almost robust in ε and quasi-optimal in the number of degrees of freedom for the balanced norm ||| · |||β.