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We are interested in the classical ill-posed Cauchy problem for the Laplace equation. One method to approximate the solution associated with compatible data consists in considering a family of regularized well-posed problems depending on a small parameter $varepsilon>0$. In this context, in order to prove convergence of finite elements methods, it is necessary to get regularity results of the solutions to these regularized problems which hold uniformly in $varepsilon$. In the present work, we obtain these results in smooth domains and in 2D polygonal geometries. In presence of corners, due the particular structure of the regularized problems, classical techniques `a la Grisvard do not work and instead, we apply the Kondratiev approach. We describe the procedure in detail to keep track of the dependence in $varepsilon$ in all the estimates. The main originality of this study lies in the fact that the limit problem is ill-posed in any framework.
The main aim of this paper is to solve an inverse source problem for a general nonlinear hyperbolic equation. Combining the quasi-reversibility method and a suitable Carleman weight function, we define a map of which fixed point is the solution to th
We study a commonly-used second-kind boundary-integral equation for solving the Helmholtz exterior Neumann problem at high frequency, where, writing $Gamma$ for the boundary of the obstacle, the relevant integral operators map $L^2(Gamma)$ to itself.
Let $H$ be a norm of ${bf R}^N$ and $H_0$ the dual norm of $H$. Denote by $Delta_H$ the Finsler-Laplace operator defined by $Delta_Hu:=mbox{div},(H( abla u) abla_xi H( abla u))$. In this paper we prove that the Finsler-Laplace operator $Delta_H$ acts
In the article a convergent numerical method for conservative solutions of the Hunter--Saxton equation is derived. The method is based on piecewise linear projections, followed by evolution along characteristics where the time step is chosen in order
This work discusses the finite element discretization of an optimal control problem for the linear wave equation with time-dependent controls of bounded variation. The main focus lies on the convergence analysis of the discretization method. The stat