30 ouvrages de mathématiques qui ont changé le monde by Jean-Jacques Samueli, Jean-Claude Boudenott

By Jean-Jacques Samueli, Jean-Claude Boudenott

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Example text

Proof. 6) arising from the bilinear form ah (u, v) is positive definite. 4) produces a unique solution. 2 Adaptivity Advection dominated problems lead to internal/boundary layers where the solution has large gradients. The standard FEMs are known to produce strong oscillations around the layers. A naive approach is to refine the mesh uniformly. But it is not desirable as it highly increase the degree of freedom and refines the mesh unnecessarily in regions where the solutions are smooth. On the other hand, in the semi-linear ADR equations, in addition to the nonphysical oscillations due to the advection, non-linear reaction can be also responsible for sharp fronts.

2: Bisection of a triangle . , let ηK1 and ηK2 be the computed local error indicators corresponding to each unknown component of a two component system. Next, we determine the set of elements MK1 and MK2 satisfying ∑ K∈MK1 (ηK1 )2 ≥ θ ∑ (ηK1 )2 , K∈ξh ∑ K∈MK2 (ηK2 )2 ≥ θ ∑ (ηK2 )2 . K∈ξh Then, we refine the marked elements K ∈ MK1 ∪MK2 using the newest vertex bisection method. The adaptive procedure ends after a sequence of mesh refinements to attain a solution within a prescribed tolerance. 11) u · ∇wdx .

2 Adaptivity 39 1/2 ∑ T3 ηJ2K |||v|||, K∈ξh which completes the proof. 4. (Bound to the conforming part of the error) The conforming part of the error satisfies u − uch dG η +Θ . 31) Proof. Since u − uch ∈ H01 (Ω ), we have |u − uch |C = |β (u − uch )|∗ . 23), we get u − uch dG = |||u − uch ||| + |u − uch |C a˜h (u − uch , v) . |||v||| v∈H 1 (Ω )\{0} sup 0 So, we need to bound the term a˜h (u − uch , v). Using the fact that u − uch ∈ H01 (Ω ), we have a˜h (u − uch , v) = a˜h (u, v) − a˜h (uch , v) = = = Ω Ω Ω f vdx − bh (u, v) − a˜h (uch , v) f vdx − bh (u, v) − Dh (uch , v) − Jh (uch , v) − Oh (uch , v) f vdx − bh (uh , v) + bh (uh , v) − bh (u, v) − a˜h (uh , v) + Dh (urh , v) + Jh (urh , v) + Oh (urh , v).

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