A fourth-order spline method for singular two-point boundary-value problems

AbstractThis paper describes two methods for the solution of (weakly) singular two-point boundary-value problems: Consider the uniform mesh xi = ih, h = 1/N, i = 0(1)N. Define the linear functionals Li(y) = y(xi) and Mi(y) = (x−α(xαy′)′\xv;x=xi. In both these methods a piecewise ‘spline’ solution is obtained in the form s(x) = si(x), x\wE; [xi−1, xi], i = 1(1)N, where in each subinterval si(x) is in the linear span of a certain set of (non-polynomial) basis functions in the representation of the solution y(x) of the two-point boundary value problem and satisfies the interpolation conditions: Li−1(s) = Li−1(y), Li(y), Mi−1(s) = Mi−1(y), Mi(s) = Mi(y). By construction s and x−α(xαs′)′ \wE; C[0,1]. Conditions of continuity are derived to ensure that xαs′ \wE; C[0, 1]. It follows that the unknown parameters yi and Mi(y), i = 1(1)N − 1, must satisfy conditions of the form: The first method consists in replacing Mi(y) by fnof(xi, yi) and solving (*) to obtain the values yi; this method is generalization of the idea of Bickley [2] for the case of (weakly) singular two-point boundary-value problems and provides order h2 uniformly convergent approximations over [0, 1]. As a modification of the above method, in the second method we generate the solution yi at the nodal points by adapting the fourth-order method of Chawla [3] and then use the conditions of continuity (*) to obtain the corresponding smoothed approximations for Mi(y) needed for the construction of the spline solution. We show that the resulting new spline method provides order h4 uniformly convergent approximations over [0, 1]. The second-order and the fourth-order methods are illustrated computationally

Optimal control for the thin-film equation: Convergence of a multi-parameter approach to track state constraints avoiding degeneracies

We consider an optimal control problem subject to the thin-film equation which is deduced from the Navier--Stokes equation. The PDE constraint lacks well-posedness for general right-hand sides due to possible degeneracies; state constraints are used to circumvent this problematic issue and to ensure well-posedness, and the rigorous derivation of necessary optimality conditions for the optimal control problem is performed. A multi-parameter regularization is considered which addresses both, the possibly degenerate term in the equation and the state constraint, and convergence is shown for vanishing regularization parameters by decoupling both effects. The fully regularized optimal control problem allows for practical simulations which are provided, including the control of a dewetting scenario, to evidence the need of the state constraint, and to motivate proper scalings of involved regularization and numerical parameters

A Modica-Mortola approximation for branched transport

The M^\alpha energy which is usually minimized in branched transport problems among singular 1-dimensional rectifiable vector measures with prescribed divergence is approximated (and convergence is proved) by means of a sequence of elliptic energies, defined on more regular vector fields. The procedure recalls the Modica-Mortola one for approximating the perimeter, and the double-well potential is replaced by a concave power

A new integral representation for quasiperiodic fields and its application to two-dimensional band structure calculations

In this paper, we consider band-structure calculations governed by the Helmholtz or Maxwell equations in piecewise homogeneous periodic materials. Methods based on boundary integral equations are natural in this context, since they discretize the interface alone and can achieve high order accuracy in complicated geometries. In order to handle the quasi-periodic conditions which are imposed on the unit cell, the free-space Green's function is typically replaced by its quasi-periodic cousin. Unfortunately, the quasi-periodic Green's function diverges for families of parameter values that correspond to resonances of the empty unit cell. Here, we bypass this problem by means of a new integral representation that relies on the free-space Green's function alone, adding auxiliary layer potentials on the boundary of the unit cell itself. An important aspect of our method is that by carefully including a few neighboring images, the densities may be kept smooth and convergence rapid. This framework results in an integral equation of the second kind, avoids spurious resonances, and achieves spectral accuracy. Because of our image structure, inclusions which intersect the unit cell walls may be handled easily and automatically. Our approach is compatible with fast-multipole acceleration, generalizes easily to three dimensions, and avoids the complication of divergent lattice sums.Comment: 25 pages, 6 figures, submitted to J. Comput. Phy