Error estimation and adaptive mesh generation in the 2D and 3D finite element method

Abstract

In order to obtain good results from the finite element method the mesh used should suit the behaviour of the field. The adaptive mesh generation offers an automatic way to generate meshes fitting the problem. In this paper the effectiveness of the adaptive mesh generation is measured by comparing the error in the field solution of uniformly and adaptively generated meshes both for the 2D and the 3D case.

Optimal discretization based adaptive finite element analysis for electromagnetics with vector tetrahedra

“…Today, the primary focus is on the research and development of effective and efficient techniques for practical 3-D applications [3]- [5]. Recent work has established the strengths of optimal discretization based refinement criteria, and confirmed the value of using functional gradient type error indicators for scalar electromagnetic applications [6], [7].…”

Probing the Numerical Convergence of a Commercial Finite Element Software in Electrochemical Simulations

“…Since the degrees of freedom (NDF) of a model is proportional to h ¹d for d-dimensional geometry, the convergence order can be defined by kek ¼ OðNDF Àq=d Þ, which is a more suitable expression for non uniform meshes. 35 The relation between NDF and the number of mesh elements (NOE) depends on the type of mesh elements and the number of dependent variables. Considering that these last parameters are kept constant throughout the work we arrive at the final expression used to define the convergence rate:…”

Computational electromagnetics for design optimisation: the state of the art and conjectures for the future

“…For example, the Incomplete Cholesky Conjugate Gradient method (ICCG) was introduced for solving large sparse systems of equations [33,34], where the operation count is approximately nlogn and largely independent of the bandwidth; the algorithm is still the root of most contemporary codes. Another significant advancement was in the now prevalent utilisation of the 'Delaunay meshing', with the original idea going back to 1934 and efficient algorithms implemented more recently in 2D [35] and 3D (using tetrahedral elements) [36] including error analysis.…”