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The general method of averaging for the superapproximation of an arbitrary partial derivative of a smooth function in a vertex $a$ of a simplicial triangulation $𝒯$ of a bounded polytopic domain in ${\Re }^d$ for any $d\ge 2$ is described and its complexity is analysed.

A method for the second-order approximation of the values of partial derivatives of an arbitrary smooth function $u=u(x_1,x_2)$ in the vertices of a conformal and nonobtuse regular triangulation ${𝒯}_h$ consisting of triangles and convex quadrilaterals is described and its accuracy is illustrated numerically. The method assumes that the interpolant ${\Pi }_h\left(u\right)$ in the finite element space of the linear triangular and bilinear quadrilateral finite elements from ${𝒯}_h$ is known only.

A straightforward generalization of a classical method of averaging is presented and its essential characteristics are discussed. The method constructs high-order approximations of the l-th partial derivatives of smooth functions u in inner vertices a of conformal simplicial triangulations T of bounded polytopic domains in ℝd for arbitrary d ≥ 2. For any k ≥ l ≥ 1, it uses the interpolants of u in the polynomial Lagrange finite element spaces of degree k on the simplices with vertex a only. The...

We study the problem of Lagrange interpolation of functions of two variables by quadratic polynomials under the condition that nodes of interpolation are vertices of a triangulation. For an extensive class of triangulations we prove that every inner vertex belongs to a local six-tuple of vertices which, used as nodes of interpolation, have the following property: For every smooth function there exists a unique quadratic Lagrange interpolation polynomial and the related local interpolation error...

A general construction of test functions in the Petrov-Galerkin method is described. Using this construction; algorithms for an approximate solution of the Dirichlet problem for the differential equation $-ϵu^n+p{u}^{\text{'}}+qu=f$ are presented and analyzed theoretically. The positive number $ϵ$ is supposed to be much less than the discretization step and the values of $\left|p\right|,q$. An algorithm for the corresponding two-dimensional problem is also suggested and results of numerical tests are introduced.

A reference triangular quadratic Lagrange finite element consists of a right triangle $\widehat{K}$ with unit legs $S_1$, $S_2$, a local space $\widehat{ℒ}$ of quadratic polynomials on $\widehat{K}$ and of parameters relating the values in the vertices and midpoints of sides of $\widehat{K}$ to every function from $\widehat{ℒ}$. Any isoparametric triangular quadratic Lagrange finite element is determined by an invertible isoparametric mapping ${ℱ}_h=(F_1,F_2)\in \widehat{ℒ}\times \widehat{ℒ}$. We explicitly describe such invertible isoparametric mappings ${ℱ}_h$ for which the images ${ℱ}_h\left(S_1\right)$, ${ℱ}_h\left(S_2\right)$ of the segments $S_1$, $S_2$ are segments,...

We analyse the error of interpolation of functions from the space $H^3(a,c)$ in the nodes $a<b<c$ of a regular quadratic Lagrange finite element in 1D by interpolants from the local function space of this finite element. We show that the order of the error depends on the way in which the mutual positions of nodes $a,b,c$ change as the length of interval $[a,c]$ approaches zero.

An explicit description of the basic Lagrange polynomials in two variables related to a six-tuple $a^1,\cdots ,a^6$ of nodes is presented. Stability of the related Lagrange interpolation is proved under the following assumption: $a^1,\cdots ,a^6$ are the vertices of triangles $T_1,\cdots ,T_4$ without obtuse inner angles such that $T_1$ has one side common with $T_j$ for $j=2,3,4$.

Let ${𝒯}_h$ be a triangulation of a bounded polygonal domain $\Omega \subset {\Re }^2$, ${ℒ}_h$ the space of the functions from $C\left(\overline{\Omega }\right)$ linear on the triangles from ${𝒯}_h$ and ${\Pi }_h$ the interpolation operator from $C\left(\overline{\Omega }\right)$ to ${ℒ}_h$. For a unit vector $z$ and an inner vertex $a$ of ${𝒯}_h$, we describe the set of vectors of coefficients such that the related linear combinations of the constant derivatives $\partial {\Pi }_h\left(u\right)/\partial z$ on the triangles surrounding $a$ are equal to $\partial u/\partial z\left(a\right)$ for all polynomials $u$ of the total degree less than or equal to two. Then we prove that, generally, the values of the...

An averaging method for the second-order approximation of the values of the gradient of an arbitrary smooth function u = u(x 1, x 2) at the vertices of a regular triangulation T h composed both of rectangles and triangles is presented. The method assumes that only the interpolant Πh[u] of u in the finite element space of the linear triangular and bilinear rectangular finite elements from T h is known. A complete analysis of this method is an extension of the complete analysis concerning the finite...

The one-dimensional steady-state convection-diffusion problem for the unknown temperature $y\left(x\right)$ of a medium entering the interval $(a,b)$ with the temperature $y_{min}$ and flowing with a positive velocity $v\left(x\right)$ is studied. The medium is being heated with an intensity corresponding to $y_{max}-y\left(x\right)$ for a constant $y_{max}>y_{min}$. We are looking for a velocity $v\left(x\right)$ with a given average such that the outflow temperature $y\left(b\right)$ is maximal and discuss the influence of the boundary condition at the point $b$ on the “maximizing” function $v\left(x\right)$.

We describe a numerical method for the equation $u_t+p{u}_x-\epsilon u_{xx}=f$ in $(0,1)\times (0,T)$ with Dirichlet boundary and initial conditions which is a combination of the method of characteristics and the finite-difference method. We prove both an a priori local error-estimate of a high order and stability. Example 3.3 indicates that our approximate solutions are disturbed only by a minimal amount of the artificial diffusion.