Advanced adaptive algorithms in 2D finite element method of higher order of accuracy

Abstract

Purpose

The paper presents the principal elements of automatic adaptivity built in our 2D software for monolithic solution of multiphysics problems based on a fully adaptive finite element method of higher order of accuracy. The adaptive techniques are illustrated by appropriate examples.

Design/methodology/approach

Presented are algorithms for realization of the h‐adaptivity, p‐adaptivity, hp‐adaptivity, creation of curvilinear elements for modelling general boundaries and interfaces. Indicated also is the possibility of combining triangular and quadrilateral elements (both classical and curved).

Findings

The presented higher‐order adaptive processes are reliable, robust and lead to a substantial reduction of the degrees of freedom in comparison with the techniques used in low‐order finite element methods. They allow solving examples that are by classical approaches either unsolvable or solvable at a cost of high memory and time of computation.

Research limitations/implications

The adaptive processes described in the paper are still limited to 2D computations. Their computer implementation is highly nontrivial (every physical field in a multiphysics task is generally solved on a different mesh satisfying its specific features) and in 3D the number of possible adaptive steps is many times higher.

Practical implications

The described adaptive techniques may represent a powerful tool for the monolithic solution of complex multiphysics problems.

Originality/value

The presented higher‐order adaptive approach of solution is shown to provide better results than the schemes implemented in professional codes based on low‐order finite element methods. Obtaining the results, moreover, requires less time and computer memory.

Citation

Karban, P., Mach, F. and Doležel, I. (2013), "Advanced adaptive algorithms in 2D finite element method of higher order of accuracy", COMPEL - The international journal for computation and mathematics in electrical and electronic engineering, Vol. 32 No. 3, pp. 834-849. https://doi.org/10.1108/03321641311305782