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An adaptive wavelet method for solving high-dimensional elliptic PDEs. (English) Zbl 1205.65313
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 \(N\).

65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
46B28 Spaces of operators; tensor products; approximation properties
65T60 Numerical methods for wavelets
35J25 Boundary value problems for second-order elliptic equations
65F35 Numerical computation of matrix norms, conditioning, scaling
65Y20 Complexity and performance of numerical algorithms
65N12 Stability and convergence of numerical methods for boundary value problems involving PDEs
Full Text: DOI
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