We propose a method for treating the Dirichlet boundary conditions in the framework of the Generalized Finite Element Method (GFEM). We use approximate Dirichlet boundary conditions as in [12] and polynomial approximations of the boundary. Our sequence of GFEM-spaces considered, Sμ, μ = 1, 2, . . . is such that Sμ 6 ⊂ H1 0 (Ω), and hence it does not conform to one of the basic FEM conditions. L...

We exhibit a large class of Lyapunov functionals for nonlinear drift–diffusion equations with non-homogeneous Dirichlet boundary conditions. These are generalizations of large deviation functionals for underlying stochastic manyparticle systems, the zero-range process and the Ginzburg–Landau dynamics, which we describe briefly. We prove, as an application, linear inequalities between such an en...

We study the exact observability of systems governed by the Schrödinger equation in a rectangle with homogeneous Dirichlet (respectively Neumann) boundary conditions and with Neumann (respectively Dirichlet) boundary observation. Generalizing results from Ramdani, Takahashi, Tenenbaum and Tucsnak (2005), we prove that these systems are exactly observable in in arbitrarily small time. Moreover, ...

In this paper, we present a priori error analysis for the finite element discretization of elliptic optimal control problems, where a finite dimensional control variable enters the Dirichlet boundary conditions. The analysis of finite element approximations of optimization problems governed by partial differential equations is an area of active research, see, e.g., [1, 12, 17, 18]. The consider...

Ranks and explicit structure of some matrices in the Quantics Tensor Train format, which allows representation with logarithmic complexity in many cases, are investigated. The matrices under consideration are Laplace operator with various boundary conditions in D dimensions and inverse Laplace operator with Dirichlet and Dirichlet-Neumann boundary conditions in one dimension. The minimal-rank e...

The author deals with the quasilinear parabolic equation ut = [uα + g(u)]∆u + buα+1+f(u,∇u) with Dirichlet boundary conditions in a bounded domain Ω, where f and g are lower order terms. He shows that, under suitable conditions on f and g, whether the solution is bounded or blows up in a finite time depends only on the first eigenvalue of −∆ in Ω with Dirichlet boundary condition. For some spec...