Advanced adaptive algorithms in 2D finite element method of higher order of accuracy
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.
automatic adaptivity higher-order finite element method convergence problems programming and algorithm theory