Abstract
Error estimates of finite element methods for reaction-diffusion problems are often realized in the related energy norm. In the singularly perturbed case, however, this norm is not adequate. A different scaling of the H1 seminorm leads to a balanced norm which reflects the layer behavior correctly.
Highlights
We shall examine the finite element method for the numerical solution of the singularly perturbed linear elliptic boundary value problemLu ≡ −ε∆u + cu = f in Ω = (0, 1) × (0, 1) (1a) u = 0 on ∂Ω,(1b) where 0 < ε 1 is a small positive parameter, c > 0 is a positive constant and f is sufficiently smooth.The problem has a unique solution u ∈ V = H01(Ω) which satisfies in the energy norm u ε := ε1/2|u|1 + u 0 f 0. (2)Here we used the following notation: if A B, there exists a constant C independent of ε such that A ≤ C B
Error estimates of finite element methods for reaction-diffusion problems are often realized in the related energy norm
When linear or bilinear elements are used on a Shishkin mesh, one can prove for the interpolation error of the Lagrange interpolant uI ∈ V N
Summary
(1b) where 0 < ε 1 is a small positive parameter, c > 0 is (for simplicity) a positive constant and f is sufficiently smooth. When linear or bilinear elements are used on a Shishkin mesh, one can prove for the interpolation error of the Lagrange interpolant uI ∈ V N u − uI ε ε1/4N −1 ln N + N −2. It follows that the error u − uN satisfies such an estimate. Error estimates in this norm are less valuable as for convection diffusion equations where the layers are of the structure exp(−x/ε). Wherefore we ask the fundamental question: Is it possible to prove error estimates in the balanced norm v b := ε1/4|v|1 + v 0 ?
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