Boundary Layer Approximate Approximations and Cubature of Potentials in Domains

Abstract: The goal of this thesis is to develop efficient numerical algorithms for approximation of functions defined on domains and apply these algorithms to obtain accurate semi-analytic cubature formulae for various integral operators. These algorithms are based on the method of "approximate approximations", introduced by V. Maz'ya in 1991. Approximate approximations are quasi-interpolants using generating functions which form an approximate partition of unity. The lack of convergence, which is not perceptible in practical computations, is compensated for by the flexibility in the choice of generating functions and the simplicity of the generalisation to the multivariate case. In the present thesis, we develop multi-resolution schemes for accurate approximation of functions up to the boundary using iterative application of quasi-interpolants on finer grids. The mesh refinement is obtained automatically through analytical factorization of the approximate approximations operator. The structure of the schemes makes it possible to perform mesh refinement locally, in particular, near the boundary. In the latter case, one obtains good accuracy on the whole domain, except on a thin boundary layer of width decreasing exponentially with the number of iterations. Using these properties we derive high-order cubature formulae for approximation of potentials acting on densities defined in domains. We give error estimates in suitable norms both for the approximation of functions and the approximation of integral operators. Special attention is paid to the construction of anisotropic schemes on affine meshes, which leads to reduced computational complexity. We conclude by deducing anisotropic formulae for approximation of the logarithmic, two-dimensional elastic and Newtonian potentials.

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