an:05659503
Zbl 1205.65313
Dijkema, Tammo Jan; Schwab, Christoph; Stevenson, Rob
An adaptive wavelet method for solving high-dimensional elliptic PDEs
EN
Constr. Approx. 30, No. 3, 423-455 (2009).
00256518
2009
j
65N30 46B28 65T60 35J25 65F35 65Y20 65N12
adaptive wavelet methods; best N-term approximations; tensor product approximation; sparse grids; matrix compression; optimal computational complexity; convergence; condition number; algorithm
Let \(\Omega=(0,1)^n\), and let \(\Gamma_D\) be the union of one or more \((n-1)\)-dimensional faces of \(\partial \Omega\). For given \(f \in (H_{0,\Gamma_D}^1(\Omega))'\), the authors study the numerical solution of the problem of finding \(u \in H_{0,\Gamma_D}^1(\Omega)\) such that
\[
a(u,v):= \int_{\Omega} c_0 uv + \sum_{m=1}^n c_m \partial_m u \; \partial_m v = f(v), \quad v \in (H_{0,\Gamma_D}^1(\Omega))',
\]
where \(c_0 \geq 0\) and \(c_m>0\), \(m=1, \ldots , n\), are constants.
The authors apply a tensor product basis \(\{\psi_{\lambda}: \lambda \in \nabla \}\) constructed by a univariate \(L^2(0,1)\)-orthonormal piecewise polynomial wavelet basis. In this case, the condition number of the stiffness matrix \(\kappa({A})\) is bounded uniformly in \(n\), and \(c_0\) and \(c_m\), \(m=1, \dots , n\). Moreover, \({A}\) is close to a sparse matrix.
The authors are interested in solutions \(u\) from the span of \(\{ \psi_{\lambda}: \lambda \in \Lambda_N \}\), where \(\Lambda_N\) is any subset with \(\#\Lambda_N=N\). They give a detailed description of an adaptive wavelet algorithm for which the resulting approximations converge in energy norm with the same rate as the best approximations from the span of the best \(N\) tensor product wavelets. Moreover, the cost for producing these approximations will be proportional to their length with a constant factor that grows only linearly with \(N\).
Gerlind Plonka-Hoch (G??ttingen)