### Superconvergence Phenomenon in the Finite Element Method Arising from Averaging Gradients.

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The solvability of time-harmonic Maxwell equations in the 3D-case with nonhomogeneous conductivities is considered by adapting Sobolev space technique and variational formulation of the problem in question. Moreover, a finite element approximation is presented in the 3D-case together with an error estimate in the energy norm. Some remarks are given to the 2D-problem arising from geophysics.

A system of first order partial differential equations is studied which is defined by the divergence and rotation operators in a bounded nonsmooth domain $\Omega \subset {\mathbf{R}}^{3}$. On the boundary $\delta \Omega $, the vanishing normal component is prescribed. A variational formulation is given and its solvability is investigated.

The authors examine a finite element method for the numerical approximation of the solution to a div-rot system with mixed boundary conditions in bounded plane domains with piecewise smooth boundary. The solvability of the system both in an infinite and finite dimensional formulation is proved. Piecewise linear element fields with pointwise boundary conditions are used and their approximation properties are studied. Numerical examples indicating the accuracy of the method are given.

The paper deals with the approximation of optimal shape of elastic bodies, unilaterally supported by a rigid, frictionless foundation. Original state inequality, describing the behaviour of such a body is replaced by a family of penalized state problems. The relation between optimal shapes for the original state inequality and those for penalized state equations is established.

We consider FE-grid optimization in elliptic unilateral boundary value problems. The criterion used in grid optimization is the total potential energy of the system. It is shown that minimization of this cost functional means a decrease of the discretization error or a better approximation of the unilateral boundary conditions. Design sensitivity analysis is given with respect to the movement of nodal points. Numerical results for the Dirichlet-Signorini problem for the Laplace equation and the...

A nonlinear elliptic partial differential equation with homogeneous Dirichlet boundary conditions is examined. The problem describes for instance a stationary heat conduction in nonlinear inhomogeneous and anisotropic media. For finite elements of degree $k\ge 1$ we prove the optimal rates of convergence $\mathcal{O}\left({h}^{k}\right)$ in the ${H}^{1}$-norm and $\mathcal{O}\left({h}^{k+1}\right)$ in the ${L}^{2}$-norm provided the true solution is sufficiently smooth. Considerations are restricted to domains with polyhedral boundaries. Numerical integration is not taken into account....

We examine new second-order necessary conditions and sufficient conditions which characterize nondominated solutions of a generalized constrained multiobjective programming problem. The vector-valued criterion function as well as constraint functions are supposed to be from the class ${C}^{1,1}$. Second-order optimality conditions for local Pareto solutions are derived as a special case.

In this note we present an algorithm for a construction of strongly regular families of triangulations for planar domains with a piecewise curved boundary. Some additional properties of the resulting triangulations are considered.

We solve a linear parabolic equation in ${\mathbb{R}}^{d}$, $d\ge 1,$ with the third nonhomogeneous boundary condition using the finite element method for discretization in space, and the $\theta $-method for discretization in time. The convergence of both, the semidiscrete approximations and the fully discretized ones, is analysed. The proofs are based on a generalization of the idea of the elliptic projection. The rate of convergence is derived also for variable time step-sizes.

We consider the finite element approximation of the identification problem, where one wishes to identify a curve along which a given solution of the boundary value problem possesses some specific property. We prove the convergence of FE-approximation and give some results of numerical tests.

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