Verifications of primal energy identities for variational problems with obstacles

We discuss error identities for two classes of free boundary problems generated by obstacles. The identities suggest true forms of the respective error measures which consist of two parts: standard energy norm and a certain nonlinear measure. The latter measure controls (in a weak sense) approximation of free boundaries. Numerical tests confirm sharpness of error identities and show that in different examples one or another part of the error measure may be dominant.Comment: 8 pages, 2 figures, conference paper: LSSC (Large-Scale scientific computing), Sozopol, Bulgaria, 2017. The final version will be published at Springe

Uniform-in-time convergence of numerical schemes for Richards' and Stefan's models

We prove that all Gradient Schemes - which include Finite Element, Mixed Finite Element, Finite Volume methods - converge uniformly in time when applied to a family of nonlinear parabolic equations which contains Richards and Stefan's models. We also provide numerical results to confirm our theoretical analysis

Towards an Efficient Finite Element Method for the Integral Fractional Laplacian on Polygonal Domains

We explore the connection between fractional order partial differential equations in two or more spatial dimensions with boundary integral operators to develop techniques that enable one to efficiently tackle the integral fractional Laplacian. In particular, we develop techniques for the treatment of the dense stiffness matrix including the computation of the entries, the efficient assembly and storage of a sparse approximation and the efficient solution of the resulting equations. The main idea consists of generalising proven techniques for the treatment of boundary integral equations to general fractional orders. Importantly, the approximation does not make any strong assumptions on the shape of the underlying domain and does not rely on any special structure of the matrix that could be exploited by fast transforms. We demonstrate the flexibility and performance of this approach in a couple of two-dimensional numerical examples

Sobolev spaces on non-Lipschitz subsets of Rn with application to boundary integral equations on fractal screens

We study properties of the classical fractional Sobolev spaces on non-Lipschitz subsets of Rn. We investigate the extent to which the properties of these spaces, and the relations between them, that hold in the well-studied case of a Lipschitz open set, generalise to non-Lipschitz cases. Our motivation is to develop the functional analytic framework in which to formulate and analyse integral equations on non-Lipschitz sets. In particular we consider an application to boundary integral equations for wave scattering by planar screens that are non-Lipschitz, including cases where the screen is fractal or has fractal boundary

Simulation of industrial crystal growth by the vertical Bridgman method

Simulation of industrial crystal growth by the vertical Bridgman method / K. G. Siebert ... - In: Mathematics - key technology for the future / Willi Jäger ... - Berlin u.a. : Springer, 2003. - S. 315-33