Veronika Sobotíková

L ∞ (L 2)-error estimates for the DGFEM applied to convection–diffusion problems on nonconforming meshes

Abstract This paper is devoted to the analysis of the discontinuous Galerkin finite element method (DGFEM) applied to the space semidiscretization of a nonlinear nonstationary convection–diffusion Dirichlet problem. General nonconforming simplicial meshes are considered and the SIPG scheme is used. Under the assumption that the exact solution is sufficiently regular an L ∞ (L 2)-optimal error estimate is derived. The theoretical results are illustrated by numerical experiments.

Numerical Integration in the DGFEM for 3D Nonlinear Convection–Diffusion Problems on Nonconforming Meshes

The effect of numerical integration in a discontinuous Galerkin finite element method for a nonstationary nonlinear convection–diffusion problem in 3D is studied. In the space semidiscretization, the volume and surface integrals are evaluated with the aid of numerical quadratures. An estimate of the error caused by the numerical integration is presented, and it is shown what quadrature formulas guarantee preservation of the accuracy of the method with exact integration.

ON THE FINITE ELEMENT ANALYSIS OF PROBLEMS WITH NONLINEAR NEWTON BOUNDARY CONDITIONS IN NONPOLYGONAL DOMAINS

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 quadratures. With the aid of some important properties of Zlamals ideal triangulation and interpolation, the convergence of the method is analyzed.

Error analysis of a DG method employing ideal elements applied to a nonlinear convection–diffusion problem

Abstract In this paper we use the discontinuous Galerkin finite element method for the space-semidiscretization of a nonlinear nonstationary convection–diffusion problem defined on a nonpolygonal two-dimensional domain. Using Zlámals concept of the ideal curved elements, we define a finite element space . We prove the ‘ideal’ versions of the inverse and the multiplicative trace inequalities known for standard straight triangulations. Further, we define a projection on the finite element space and study its approximation properties. The obtained results allow us to derive an H 1-optimal error estimate for the discontinuous Galerkin method employing the ideal curved elements.

An Error Estimate for the Finite Element Solution of an Elliptic Problem with a Nonlinear Newton Boundary Condition in Nonpolygonal Domains

Abstract The article is concerned with the study of an elliptic boundary value problem with a nonlinear Newton boundary condition considered in a two-dimensional domain with a curved boundary. The existence and uniqueness of the weak solution of the continuous problem is a consequence of the monotone operator theory. The problem is discretized with the use of the finite element method. The main attention is paid to the effect of the approximation of the curved boundary by a piecewise linear boundary and of the evaluation of integrals by numerical quadratures. With the aid of some important properties of Zlámals ideal triangulation and interpolation, the error estimate for the solution of the discrete problem is derived.

Ideal Curved Elements and the Discontinuous Galerkin Method

In this paper we prove a new result concerning Zlamal’s ideal curved elements which allows us to employ these elements in a discontinuous Galerkin finite element method for a nonlinear convection-diffusion problem on a nonpolygonal domain, and to derive an H 1-optimal error estimate for this method.

Numerical Integration in the Discontinuous Galerkin Method for Nonlinear Convection-Diffusion Problems in 3D

In this paper the discontinuous Galerkin finite element method is used for the space-semidiscretization of a nonlinear nonstationary convection-diffusion problem in three dimensions. As in practical computations integrals appearing in the forms defining the approximate solution are evaluated with the use of quadrature formulae, the effect of numerical integration in the method is studied. An estimate of the error caused by the numerical integration is presented and it is shown which quadrature formulae guarantee preservation of the accuracy of the method with exact integration.

Numerical integration in the DGFEM for nonlinear convection-diffusion problems

The effect of numerical integration in the DGFEM for nonlinear convection-diffusion problems in 2D is studied. The volume and line integrals in the space semidiscretization are evaluated by numerical quadratures. The main goal is to estimate the error caused by the numerical integration and to show what numerical quadratures should be used in order to preserve the accuracy of the method with exact integration.

The Finite Element Analysis of an Elliptic Problem with a Nonlinear Newton Boundary Condition

Results of the study of an elliptic 2D problem with a nonlinear Newton boundary condition are presented. The problem is discretized with the use of the FEM and the integrals are evaluated by numerical quadratures. In the case of a nonpolygonal domain the main attention is paid to the effect of a piecewise linear approximation of the boundary. The error estimate for the solution of the discrete FE problem is derived.