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On the effect of numerical integration in the finite element solution of an elliptic problem with a nonlinear Newton boundary condition

Ondřej Bartoš, Miloslav Feistauer, Filip Roskovec (2019)

Applications of Mathematics

This paper is concerned with the analysis of the finite element method for the numerical solution of an elliptic boundary value problem with a nonlinear Newton boundary condition in a two-dimensional polygonal domain. The weak solution loses regularity in a neighbourhood of boundary singularities, which may be at corners or at roots of the weak solution on edges. The main attention is paid to the study of error estimates. It turns out that the order of convergence is not dampened by the nonlinearity...

On the efficient use of the Galerkin-method to solve Fredholm integral equations

Wolfgang Hackbusch, Stefan A. Sauter (1993)

Applications of Mathematics

In the present paper we describe, how to use the Galerkin-method efficiently in solving boundary integral equations. In the first part we show how the elements of the system matrix can be computed in a reasonable time by using suitable coordinate transformations. These techniques can be applied to a wide class of integral equations (including hypersingular kernels) on piecewise smooth surfaces in 3-D, approximated by spline functions of arbitrary degree. In the second part we show, how to use the...

On the existence of a weak solution of the boundary value problem for the equilibrium of a shallow shell reinforced with stiffening ribs

Igor Bock, Ján Lovíšek (1978)

Aplikace matematiky

The existence and the unicity of a weak solution of the boundary value problem for a shallow shell reinforced with stiffening ribs is proved by the direct variational method. The boundary value problem is solved in the space W ( Ω ) ⊂ H 0 1 ( Ω ) × H 0 1 ( Ω ) × H 0 2 ( Ω ) , on which the corresponding bilinear form is coercive. A finite element method for numerical solution is introduced. The approximate solutions converge to a weak solution in the space Q ( Ω ) .

On the finite element analysis of problems with nonlinear Newton boundary conditions in nonpolygonal domains

Miloslav Feistauer, Karel Najzar, Veronika Sobotíková (2001)

Applications of Mathematics

The paper is concerned with the study of an elliptic boundary value problem with a nonlinear Newton boundary condition considered in a two-dimensional nonpolygonal domain with a curved boundary. The existence and uniqueness of the solution of the continuous problem is a consequence of the monotone operator theory. The main attention is paid to the effect of the basic finite element variational crimes: approximation of the curved boundary by a polygonal one and the evaluation of integrals by numerical...

On the inf-sup condition for higher order mixed FEM on meshes with hanging nodes

Vincent Heuveline, Friedhelm Schieweck (2007)

ESAIM: Mathematical Modelling and Numerical Analysis

We consider higher order mixed finite element methods for the incompressible Stokes or Navier-Stokes equations with Qr-elements for the velocity and discontinuous P r - 1 -elements for the pressure where the order r can vary from element to element between 2 and a fixed bound r * . We prove the inf-sup condition uniformly with respect to the meshwidth h on general quadrilateral and hexahedral meshes with hanging nodes.

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