The aim of this paper is to develop a functional-analytic framework for the construction of level set methods, when applied to shape optimization and shape reconstruction problems. As a main tool we use a notion of gradient ows for geometric configurations such as used in the modelling of geometric motions in materials science. The analogies to this field lead to a scale of level set evolutions, characterized by the norm used for the choice of the velocity. This scale of methods also includes the standard approach used in previous work on this subject as a special case.

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Discretization of singular functions is an important component in many problems to which level set methods have been applied. We present two methods for constructing consistent approximations to Dirac delta measures concentrated on piecewise smooth curves or surfaces. Both methods are designed to be convenient for level set simulations on Cartesian grids and are introduced to replace the commonly used but inconsistent regularization technique that is solely based on the distance to the singularity with a regularization parameter proportional to the mesh size. The first algorithm is based on a tensor product of regularized one-dimensional delta functions.

Singular terms in di#erential equations pose severe challenges for numerical approximations on regular grids. Regularization of the singularities is a very useful technique for their representation on the grid. We analyze such techniques for the practically preferred case of narrow support of the regularizations, extending our earlier results for wider support. The analysis also generalizes existing theory for one dimensional problems to multi dimensions. New high order multi dimensional techniques for di#erential equations and numerical quadrature are introduced based on the analysis and numerical results are presented. We also show that the common use of distance functions to extend one dimensional regularization to higher dimensions may produce O(1) errors.

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An International Journal computers &

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www.elsevier.com/locate/jcp Numerical approximations of singular source terms in differential equations

10 The pointwise estimation of heat conduction solution as a function of truncation error of a finite difference scheme is 11 addressed. The truncation error is estimated using a Taylor series with the remainder in the Lagrange form. The con-12 tribution of the local error to the total pointwise error is estimated via an adjoint temperature. It is demonstrated that 13 the results of numerical calculation of the temperature at an observation point may thus be refined via adjoint error 14 correction and that an asymptotic error bound may be found.

A posteriori pointwise error estimation for compressible uid ows using adjoint parameters and Lagrange remainder 5 7