### A general theorem on triangular finite ${C}^{\left(m\right)}$-elements

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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...

Making use of a line integral defined without use of the partition of unity, Green’s theorem is proved in the case of two-dimensional domains with a Lipschitz-continuous boundary for functions belonging to the Sobolev spaces ${W}^{1,p}\left(\right)\equiv {H}^{1,p}\left(\right)$ $(1\le p<)$.

Extensions from ${H}^{1}\left({\Omega}_{P}\right)$ into ${H}^{1}\left(\Omega \right)$ (where ${\Omega}_{P}\subset \Omega $) are constructed in such a way that extended functions satisfy prescribed boundary conditions on the boundary $\partial \Omega $ of $\Omega $. The corresponding extension operator is linear and bounded.

A modification of the Nikolskij extension theorem for functions from Sobolev spaces ${H}^{k}\left(\Omega \right)$ is presented. This modification requires the boundary $\partial \Omega $ to be only Lipschitz continuous for an arbitrary $k\in \mathbb{N}$; however, it is restricted to the case of two-dimensional bounded domains.

Making use of a surface integral defined without use of the partition of unity, trace theorems and the Gauss-Ostrogradskij theorem are proved in the case of three-dimensional domains $\Omega $ with a Lipschitz-continuous boundary for functions belonging to the Sobolev spaces ${H}^{1,p}\left(\right)$ $(1\le p<)$. The paper is a generalization of the previous author’s paper which is devoted to the line integral.

Approximation of nonhomogeneous boundary conditions of Dirichlet and Neumann types is suggested in solving boundary value problems of elliptic equations by the finite element method. Curved triangular elements are considered. In the first part of the paper the convergence of the finite element method is analyzed in the case of nonhomogeneous Dirichlet problem for elliptic equations of order $2m+2$, in the second part of the paper in the case of nonhomogeneous mixed boundary value problem for second order...

Existence and uniqueness theorem is established for a variational problem including Biot's model of consolidation of clay. The proof of existence is constructive and uses the compactness method. Error estimates for the approximate solution obtained by a method combining finite elements and Euler's backward method are given.

The finite element analysis of linear elliptic problems in two-dimensional domains with cusp points (turning points) is presented. This analysis needs on one side a generalization of results concerning the existence and uniqueness of the solution of a constinuous elliptic variational problem in a domain the boundary of which is Lipschitz continuous and on the other side a presentation of a new finite element interpolation theorem and other new devices.

The paper deals with an initial problem of a parabolic variational inequality whichcontains a nonlinear elliptic form $a(v,w)$ having a potential $J\left(v\right)$, which is twice $G$-differentiable at arbitrary $v\in {H}^{1}\left(\Omega \right)$. This property of $a(v,w)$ makes it possible to prove convergence of an approximate solution defined by a linearized scheme which is fully discretized - in space by the finite elements method and in time by a one-step finite-difference method. Strong convergence of the approximate solution is proved without any regularity...

Curved triangular ${C}^{m}$-elements which can be pieced together with the generalized Bell’s ${C}^{m}$-elements are constructed. They are applied to solving the Dirichlet problem of an elliptic equation of the order $2(m+1)$ in a domain with a smooth boundary by the finite element method. The effect of numerical integration is studied, sufficient conditions for the existence and uniqueness of the approximate solution are presented and the rate of convergence is estimated. The rate of convergence is the same as in the...

The finite element method is a generalized Ritz method using special admissible functions. In the paper, triangular elements and functions are considered which are linear or quadratic polynomials on each triangle. The convergence is proved for variational problems arising from second order boundary value problems. The order of accuracy of the procedure is $(s+1)/2$ in case of inhomogeneous Dirichlet conditions and $s$ in other cases ($s$ is the degree of the polynomial used).

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