\(L^{\infty}\)-stable approximation of a solution to \(\mathrm{div}(Y)= f\) for \(f \in L^2\) in two dimensions. (English) Zbl 1108.65107
Given \(f\in L^2\) on the torus in \(2\)-space, a solution \(Y\) of \(\operatorname{div}Y=F\) which is stable in the \(L^\infty\) norm cannot be constructed by a linear operator. This was proven by J. Bourgain and H. Brezis [J. Am. Math. Soc. 16, 393–426 (2002; Zbl 1075.35006)]. Therefore an approximation is considered here which is based on truncated Fourier series and a nonlinear optimization procedure. The essential tools are relations between \(L^\infty\)-norms, \(H^1\)-norms, and \(L^p\)-norms \((p=p(N))\) of trigonometric polynomials.
Reviewer: Dietrich Braess (Bochum)
MSC:
65N12 | Stability and convergence of numerical methods for boundary value problems involving PDEs |
35F15 | Boundary value problems for linear first-order PDEs |
65N22 | Numerical solution of discretized equations for boundary value problems involving PDEs |
Keywords:
inversion of divergence; stabilith divergence equations; critical Sobolev norms; Fourier approximation; discrete solution; Fourier orthogonal projection operatorCitations:
Zbl 1075.35006References:
[1] | Bourgain J., Brezis H. (2002). On the equation div(Y) =f and application to control of phases. J. A.M.S. 16(2): 393–426 · Zbl 1075.35006 |
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