Pavel Šolín

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

Arbitrary-level hanging nodes and automatic adaptivity in the hp-FEM

In this paper we present a new automatic adaptivity algorithm for the hp-FEM which is based on arbitrary-level hanging nodes and local element projections. The method is very simple to implement compared to other existing hp-adaptive strategies, while its performance is comparable or superior. This is demonstrated on several numerical examples which include the L-shape domain problem, a problem with internal layer, and the Girkmann problem of linear elasticity. With appropriate simplifications, the proposed technique can be applied to standard lower-order and spectral finite element methods.

Modular hp-FEM system HERMES and its application to Maxwell's equations

In this paper, we introduce a multi-physics modular hp-FEM system HERMES. The code is based on a novel approach where the finite element technology (mesh processing and adaptation, numerical quadrature, assembling and solution of the discrete problems, a-posteriori error estimation, etc.) is fully separated from the physics of the solved problems. The physics is represented via simple modules containing PDE-dependent parameters as well as hierarchic higher-order finite elements satisfying the conformity requirements imposed by the PDE. After describing briefly the modular structure of HERMES and some of its functionality, we focus on its application to the time-harmonic Maxwells equations. We present numerical results which illustrate the capability of the hp-FEM to reduce both the number of degrees of freedom and the CPU time dramatically compared to standard lowest-order FEM.

Adaptive higher-order finite element methods for transient PDE problems based on embedded higher-order implicit Runge-Kutta methods

We present a new class of adaptivity algorithms for time-dependent partial differential equations (PDE) that combine adaptive higher-order finite elements (hp-FEM) in space with arbitrary (embedded, higher-order, implicit) Runge-Kutta methods in time. Weak formulation is only created for the stationary residual, and the Runge-Kutta methods are specified via their Butchers tables. Around 30 Butchers tables for various Runge-Kutta methods with numerically verified orders of local and global truncation errors are provided. A time-dependent benchmark problem with known exact solution that contains a sharp moving front is introduced, and it is used to compare the quality of seven embedded implicit higher-order Runge-Kutta methods. Numerical experiments also include a comparison of adaptive low-order FEM and hp-FEM with dynamically changing meshes. All numerical results presented in this paper were obtained using the open source library Hermes (http://www.hpfem.org/hermes) and they are reproducible in the Networked Computing Laboratory (NCLab) at http://www.nclab.com.

Imposing orthogonality to hierarchic higher-order finite elements

We propose a new class of hierarchic higher-order finite elements suitable for the hp-FEM discretization of symmetric linear elliptic problems. These elements use shape functions which are partially orthonormal on the reference domain under the energetic inner product induced by the elliptic problem. We present numerical experiments showing excellent conditioning properties of the new partially orthogonal shape functions compared to other popular sets of hierarchic shape functions.

Discrete maximum principle for a 1D problem with piecewise-constant coefficients solved by hp-FEM

In this paper we prove the discrete maximum principle for a one-dimensional equation of the form –(au′)′ = f with piecewise-constant coefficient a(x), discretized by the hp-FEM. The discrete problem is transformed in such a way that the discontinuity of the coefficient a(x) disappears. Existing results are then applied to obtain a condition on the mesh which guarantees the satisfaction of the discrete maximum principle. Both Dirichlet and mixed Dirichlet–Neumann boundary conditions are discussed.

Discrete conservation of nonnegativity for elliptic problems solved by the hp-FEM

Most results related to discrete nonnegativity conservation principles (DNCP) for elliptic problems are limited to finite differences and lowest-order finite element methods (FEM). In this paper we show that a straightforward extension to higher-order finite element methods (hp-FEM) in the classical sense is not possible. We formulate a weaker DNCP for the Poisson equation in one spatial dimension and prove it using an interval computing technique. Numerical experiments related to the extension of this result to 2D are presented.

Computation of general nonstationary 2D eddy currents in linear moving arrangements using integrodifferential approach

Purpose – Most eddy current problems are solved using numerical schemes based on the differential approach. Nevertheless, there exist several classes of tasks where use of this approach may be complicated (problems characterized by geometrical incommensurability of individual subdomains, motion, etc.). In such cases, application of the integrodifferential approach may be an advantage. The paper seeks to present the theoretical background of the method.Design/methodology/approach – The mathematical model consists of a system of integrodifferential equations for current densities in electrically conductive domains.Findings – The methodology is illustrated on an example. All computations are realized by a code developed and written by the authors.Originality/value – The presented algorithm based on the integrodifferential approach makes it possible to solve problems that are only hardly solvable by classical differential methods.

Solution of 3D singular electrostatics problems using adaptive hp‐FEM

Purpose – This paper seeks to describe the solution of a simple electrostatic problem using an adaptive hp‐FEM and to show the benefits of this approach. Numerical experiments are presented to demonstrate its superiority.Design/methodology/approach – Adaptive hp‐FEM is used. In contrast with standard FEM, the automatic adaptivity procedure can choose from a variety of refinement candidates. An element with over estimated error can be refined in space, or its polynomial degree can be increased. Arbitrary level hanging nodes are allowed, so that no unnecessary refinements are performed in order to keep a mesh regular.Findings – Numerical solution of a singular electrostatic problem is presented. From the comparison it can be seen that the hp‐FEM outperforms both the standard linear and quadratic elements significantly. The accuracy of an hp‐FEM solution would be hard to attain by standard means due to the limited capacity of the computer memory.Originality/value – The paper describes results obtained from a...

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.