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For compact domains with smooth boundaries, we present a surface spline approximation scheme that delivers rates in Lp that are optimal for linear approximation in this setting. This scheme can overcome the boundary effects, observed by Johnson [Constr. Approx., 14 (1998), pp. 429--438], by placing centers with greater density near the boundary. It owes its success to an integral identity employing a minimal number of boundary layer potentials, which, in turn, is derived from the boundary layer potential solution to the Dirichlet problem for the m-fold Laplacian. Furthermore, this integral identity is shown to be the "native space extension" of the target function. [ABSTRACT FROM AUTHOR]

Anisotropically refined mixed finite elements are beneficial for the resolution of local features such as boundary layers. Unfortunately, the stability of the resulting scheme is highly sensitive to the aspect ratio of the elements. Previous analysis revealed that the degeneration arises from a relatively small number of spurious (piecewise constant) pressure modes. The present article is concerned with resolving the problem of how to suppress the spurious pressure modes in order to restore stability yet at the same time not incur any deterioration in the approximation properties of the reduced pressure space. Two results are presented. The first gives the minimal constraints on the pressure space needed to restore stability with respect to aspect ratio and shows that the approximation properties of the constrained pressure space and the unconstrained pressure space are essentially identical. Alternatively, one can impose the constraint weakly through the use of a stabilized finite element scheme. A second result shows that the stabilized finite element scheme is robust with respect to the aspect ratio of the elements and produces an approximation that satisfies an error bound of the same type to the mixed finite element scheme using the constrained space. [ABSTRACT FROM AUTHOR]

We use a recently developed method [O. Costin, M. Huang, and W. Schlag, Nonlin-earity, 25 (2012), pp. 125-164], [O. Costin, M. Huang, and S. Tanveer, Duke Math. J., 163 (2014), pp. 665-704] to find accurate analytic approximations with rigorous error bounds for the classic similarity solution of Blasius of the boundary layer equation in fluid mechanics, the two-point boundary value problem f'" + f f" = 0 with f(0) = f'(0) = 0 and limx→∞ f'(x) = 1. The approximation is given in terms of a polynomial in [0,5/2] and in terms of the error function in [5/2, ∞). The two representations for the solution in different domains match at x = 5/2 determining all free parameters in the problem, in particular f"(0) = 0.469600 ± 0.000022 at the wall. The method can in principle provide approximations to any desired accuracy for this or wide classes of linear or nonlinear differential equations with initial or boundary value conditions. The analysis relies on controlling the errors in the approximation through contraction mapping arguments, using energy bounds for the Green's function of the linearized problem. [ABSTRACT FROM AUTHOR]