Karel Segeth

Higher-Order Finite Element Methods

INTRODUCTION Finite Elements Orthogonal Polynomials A One-Dimensional Example HIERARCHIC MASTER ELEMENTS OF ARBITRARY ORDER De Rham Diagram H^1-Conforming Approximations H(curl)-Conforming Approximations H(div)-Conforming Approximations L^2-Conforming Approximations HIGHER-ORDER FINITE ELEMENT DISCRETIZATION Projection-Based Interpolation on Reference Domains Transfinite Interpolation Revisited Construction of Reference Maps Projection-Based Interpolation on Physical Mesh Elements Technology of Discretization in Two and Three Dimensions Constrained Approximation Selected Software-Technical Aspects HIGHER-ORDER NUMERICAL QUADRATURE One-Dimensional Reference Domain K(a) Reference Quadrilateral K(q) Reference Triangle K(t) Reference Brick K(B) Reference Tetrahedron K(T) Reference Prism K(P) NUMERICAL SOLUTION OF FINITE ELEMENT EQUATIONS Direct Methods for Linear Algebraic Equations Iterative Methods for Linear Algebraic Equations Choice of the Method Solving Initial Value Problems for ordinary Differential Equations MESH OPTIMIZATION, REFERENCE SOLUTIONS, AND hp-ADAPTIVITY Automatic Mesh Optimization in One Dimension Adaptive Strategies Based on Automatic Mesh Optimization Goal-Oriented Adaptivity Automatic Goal-Oriented h-, p-, and hp-Adaptivity Automatic Goal-Oriented hp-Adaptivity in Two Dimensions

Three-dimensional hydrodynamical modelling of viscous flow around a rotating ellipsoidial inclusion

Abstract A procedure and a FORTRAN program are presented which enable numerical modelling of viscous flow around a rotating rigid ellipsoidal inclusion. For any homogeneous flow given by a velocity gradient tensor it is possible to trace the rotation of the inclusion, velocity field around it, and also the motion of a chosen set of passive markers within the fluid. Examples of modelling were visualized using the MATLAB Package.

A review of some a posteriori error estimates for adaptive finite element methods

Recently, the adaptive finite element methods have gained a very important position among numerical procedures for solving ordinary as well as partial differential equations arising from various technical applications. While the classical a posteriori error estimates are oriented to the use in h-methods the contemporary higher order hp-methods usually require new approaches in a posteriori error estimation. We present a brief review of some error estimation procedures for some particular both linear and nonlinear differential problems with special regards to the needs of the hp-method.

Hierarchic higher-order hermite elements on hybrid triangular/quadrilateral meshes

In this paper we propose a new family of hierarchic higher-order Hermite elements on hybrid triangular/quadrilateral meshes. Optimal higher-order interior modes are calculated using an appropriate generalized eigenvalue problem. New nonaffine reference mappings preserving the continuity of derivatives at grid vertices are developed.

Higher-Order FEM for a System of Nonlinear Parabolic PDE’s in 2D with A-Posteriori Error Estimates

Initial-boundary value problems for systems of nonlinear parabolic partial differential equations arise in many important practical applications in electromagnetics, chemistry, modelling of diffusion and heat transfer processes and other fields. We are concerned with their solution by means of the method of lines with higher-order finite element spatial discretization on unstructured triangular meshes. Obviously, development of realistic a-posteriori error estimates plays an essential role in the application of a strategy of this type.

A new sequence of hierarchic prismatic elements satisfying De Rham diagram on hybrid meshes

This paper presents a new sequence of affine-equivalent H(curl)- and H(div)-conforming hierarchic prismatic finite elements of arbitrary polynomial degrees, satisfying De Rham diagram on hybrid tetrahedral-prismatic-hexahedral meshes. We also present suitable H(curl)- and H(div)-conforming reference maps that preserve the commutativity of the De Rham diagram.

Examples of non-uniqueness of almost-unidirectional gas flow

Modeling of non-linear phenomena always brings the question of uniqueness of the exact as well as numerical solution. The paper is devoted to the non-unique behavior of gases, which has been observed in special axisymmetric nozzles at transonic flow regimes. We describe the flow by means of the compressible Euler equations. Surprisingly, the non-uniqueness is recorded already by the simplest stationary quasi-one-dimensional model. In this case, we prove the non-uniqueness analytically. The non-unique behavior is present also in the three-dimensional model, which we illustrate numerically using an original mass-conserving axisymmetric finite volume scheme. Numerical example of non-unique solutions corresponding to an axisymmetric nozzle of a complicated geometry shows a good agreement between the quasi-one-dimensional analytical and axisymmetric three-dimensional numerical results.

Description of the Multi-Dimensional Finite Volume Solver EULER

This paper is aimed at the description of the multi-dimensional finite volume solver EULER, which has been developed for the numerical solution of the compressible Euler equations during several last years. The present overview of numerical schemes and the explanation of numerical techniques and tricks which have been used for EULER could be of certain interest not only for registered users but also for numerical mathematicians who have decided to implement a finite volume solver themselves. This solver has been used also for the computation of numerical examples presented in other papers of the authors.