Nonhomogeneous Neumann Boundary Research Articles

Backward Euler method for abstract time-dependent parabolic equations with variable domains

We consider semidiscretizations in time, based on the backward Euler method, of an abstract, non-autonomous parabolic initial value problem \(\) where \(A(t) : D(A(t)) \subset X \to X\), \(0 \le t \le T\), is a family of sectorial operators in a Banach space X. The domains \(D(A(t))\) are allowed to depend on t. Our hypotheses are fulfilled for classical parabolic problems in the \(L^p\), \(1 < p <+\infty\), norms. We prove that the semidiscretization is stable in a suitable sense. We get optimal estimates for the error even when non-homogeneous boundary values are considered. In particular, the results are applicable to the analysis of the semidiscretizations of time-dependent parabolic problems under non-homogeneous Neumann boundary conditions.

Finite element variational crimes in the case of semiregular elements

The finite element method for a strongly elliptic mixed boundary value problem is analyzed in the domain $\Omega $ whose boundary $\partial \Omega $ is formed by two circles $\Gamma _1$, $\Gamma _2$ with the same center $S_0$ and radii $R_1$, $R_2=R_1+\varrho $, where $\varrho \ll R_1$. On one circle the homogeneous Dirichlet boundary condition and on the other one the nonhomogeneous Neumann boundary condition are prescribed. Both possibilities for $u=0$ are considered. The standard finite elements satisfying the minimum angle condition are in this case inconvenient; thus triangles obeying only the maximum angle condition and narrow quadrilaterals are used. The restrictions of test functions on triangles are linear functions while on quadrilaterals they are four-node isoparametric functions. Both the effect of numerical integration and that of approximation of the boundary are analyzed. The rate of convergence $O(h)$ in the norm of the Sobolev space $H^1$ is proved under the following conditions: 1. the

Convergence of optimal solutions in control problems for hyperbolic equations

A sequence of optimal control problems for systems governed by linear hyperbolic equations with the nonhomogeneous Neumann boundary conditions is considered. The integral cost functionals and the differential operators in the equations depend on the param

Use of Ritz and Bubnow-Galerkin methods for calculation of inductance and impedance of conductors

It is shown that the Ritz and Bubnow-Galerkin methods can be applied for the calculation of the inductance and the impedance, respectively. Moreover it is shown that the calculation of the impedance by the Bubnow-Galerkin method is a generalization of the calculation of the inductance by the Ritz method in the case of a nonhomogeneous Neumann boundary problem. For the illustration of these methods the inductance of a rectangular ferromagnetic conductor and the inner impedance of a rectangular conductor placed in a semi-closed slot are calculated.