Nonsmooth Boundary Data Research Articles

Second order discretization of backward SDEs and simulation with the cubature method

We propose a second order discretization for backward stochastic differential equations (BSDEs) with possibly nonsmooth boundary data. When implemented, the discretization method requires essentially the same computational effort with the Euler scheme for BSDEs of Bouchard and Touzi [Stochastic Process. Appl. 111 (2004) 175–206] and Zhang [Ann. Appl. Probab. 14 (2004) 459–488]. However, it enjoys a second order asymptotic rate of convergence, provided that the coefficients of the equation are sufficiently smooth. In the second part of the paper, we combine this discretization with higher order cubature formulas on Wiener space to produce a fully implementable second order scheme.

A time-marching MFS scheme for heat conduction problems

In this work we consider the numerical solution of a heat conduction problem for a material with non-constant properties. By approximating the time derivative of the solution through a finite difference, the transient equation is transformed into a sequence of inhomogeneous Helmholtz-type equations. The corresponding elliptic boundary value problems are then solved numerically by a meshfree method using fundamental solutions of the Helmholtz equation as shape functions. Convergence and stability of the method are addressed. Some of the advantages of this scheme are the absence of domain or boundary discretizations and/or integrations. Also, no auxiliary analytical or numerical methods are required for the derivation of the particular solution of the inhomogeneous elliptic problems. Numerical simulations for 2D domains are presented. Smooth and non-smooth boundary data will be considered.

The Boussinesq system with mixed nonsmooth boundary data

We treat the stationary Boussinesq system with nonsmooth mixed boundary conditions for the temperature, and nonsmooth Dirichlet boundary condition for the velocity. We prove the existence, the continuous dependence of the solution with respect to the data and the uniqueness of the very weak solution. To cite this article: E.J. Villamizar-Roa et al., C. R. Acad. Sci. Paris, Ser. I 343 (2006).

EXISTENCE AND REGULARITY RESULTS FOR THE STOKES SYSTEM WITH NON-SMOOTH BOUNDARY DATA IN A POLYGON

We discuss the problem of existence and uniqueness of solutions for the nonhomogeneous Stokes system in a plane polygon with non-smooth boundary data.

Fully discrete approximations of parabolic boundary-value problems with nonsmooth boundary data

Fully discrete approximations of parabolic boundary-value problems with nonsmooth boundary data

Optimal convergence rates for semidiscrete approximations of parabolic problems with nonsmooth boundary data

We consider semidiscrete approximations of parabolic boundary value problems based on an elliptic approximation by J. Nitsche, in which the approximating subspaces are not subject to any boundary conditions. Optimal Lp (L 2) error estimates are derived for both smooth and nonsmooth boundary data. The approach is based on semigroup theory combined with the theory of singular integrals.

Fully discrete Galerkin approximations of parabolic boundary-value problems with nonsmooth boundary data

In this paper we study the convergence properties of a fully discrete Galerkin approximation with a backwark Euler time discretization scheme. An approach based on semigroup theory is used to deal with the nonsmooth Dirichlet boundary data which cannot be handled by standard techniques. This approach gives rise to optimal rates of convergence inL p[O,T;L 2(Ω)] norms for boundary conditions inL p[O,T;L 2(?)], 1?p??.

Semidiscrete Approximations of Hyperbolic Boundary Value Problems with Nonhomogeneous Dirichlet Boundary Conditions

Finite-element approximations of the wave equation with nonhomogenous and “nonsmooth” Dirichlet boundary data are considered. These approximations are based on a special Variational regularization of the problem introduced by J. L. Lions. The convergence rates of the approximations with nonsmooth boundary data are derived.