On Patched Variational Principles with Application to Elliptic and Mixed Elliptic-Hyperbolic Problems.
We prove the phase segregation phenomenon to occur in the ground state solutions of an interacting system of two self-coupled repulsive Hartree equations for large nonlinear and nonlocal interactions. A self-consistent numerical investigation visualizes the approach to this segregated regime.
We deal with the numerical solution of elliptic not necessarily self-adjoint problems. We derive a posteriori upper bound based on the flux reconstruction that can be directly and cheaply evaluated from the original fluxes and we show for one-dimensional problems that local efficiency of the resulting a posteriori error estimators depends on only, where is the discretization polynomial degree. The theoretical results are verified by numerical experiments.
We consider triangulations formed by triangular elements. For the standard linear interpolation operator we prove the interpolation order to be for provided the corresponding family of triangulations is only semiregular. In such a case the well-known Zlámal’s condition upon the minimum angle need not be satisfied.
In this paper we present theoretical, computational, and practical aspects concerning 3-dimensional shape optimization governed by linear magnetostatics. The state solution is approximated by the finite element method using Nédélec elements on tetrahedra. Concerning optimization, the shape controls the interface between the air and the ferromagnetic parts while the whole domain is fixed. We prove the existence of an optimal shape. Then we state a finite element approximation to the optimization...
It is well known that finite element spaces used for approximating the velocity and the pressure in an incompressible flow problem have to be stable in the sense of the inf-sup condition of Babuška and Brezzi if a stabilization of the incompressibility constraint is not applied. In this paper we consider a recently introduced class of triangular nonconforming finite elements of th order accuracy in the energy norm called elements. For we show that the stability condition holds if the velocity...
We are interested in the numerical solution of a two-dimensional fluid-structure interaction problem. A special attention is paid to the choice of physically relevant inlet boundary conditions for the case of channel closing. Three types of the inlet boundary conditions are considered. Beside the classical Dirichlet and the do-nothing boundary conditions also a generalized boundary condition motivated by the penalization prescription of the Dirichlet boundary condition is applied. The fluid flow...
The maximum angle condition of J. L. Synge was originally introduced in interpolation theory and further used in finite element analysis and applications for triangular and later also for tetrahedral finite element meshes. In this paper we present some of its generalizations to higher-dimensional simplicial elements. In particular, we prove optimal interpolation properties of linear simplicial elements in that degenerate in some way.
A new approach for computationally efficient estimation of stability factors for parametric partial differential equations is presented. The general parametric bilinear form of the problem is approximated by two affinely parametrized bilinear forms at different levels of accuracy (after an empirical interpolation procedure). The successive constraint method is applied on the coarse level to obtain a lower bound for the stability factors, and this bound is extended to the fine level by adding a proper...
A new approach for computationally efficient estimation of stability factors for parametric partial differential equations is presented. The general parametric bilinear form of the problem is approximated by two affinely parametrized bilinear forms at different levels of accuracy (after an empirical interpolation procedure). The successive constraint method is applied on the coarse level to obtain a lower bound for the stability factors, and this...
A new approach for computationally efficient estimation of stability factors for parametric partial differential equations is presented. The general parametric bilinear form of the problem is approximated by two affinely parametrized bilinear forms at different levels of accuracy (after an empirical interpolation procedure). The successive constraint method is applied on the coarse level to obtain a lower bound for the stability factors, and this...