Artificial Boundary Conditions for the Stokes and Navier–Stokes Equations in Domains that are Layer-Like at Infinity

Artificial boundary conditions are presented to approximate solutions to Stokesand Navier-Stokes problems in domains that are layer-like at infinity. Based on results about existence and asymptotics of the solutions v∞, p∞ to the problems in the unbounded domain Ω the error v∞−vR, p∞−pR is estimated in H(ΩR) and L(ΩR), respectively. Here v, p are the approximating solutions on the truncated domain ΩR, the parameter R controls the exhausting of Ω. The artificial boundary conditions involve the Steklov-Poincare operator on a circle together with its inverse and thus turn out to be a combination of local and nonlocal boundary operators. Depending on the asymptotic decay of the data of the problems, in the linear case the error vanishes of order O(R−N ), where N can be arbitrarily large.

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