Analysis of Finite Element Domain Embedding Methods for Curved Domains using Uniform Grids

2008 ◽  
Vol 46 (6) ◽  
pp. 2843-2866 ◽  
Author(s):  
Sheng Zhang
Keyword(s):  
Finite Element ◽  
Uniform Grids ◽  
Curved Domains ◽  
Embedding Methods ◽  
Domain Embedding Methods

scholarly journals Error indicators and adaptive refinement of the discrete thin plate spline smoother

2019 ◽  
Vol 60 ◽  
pp. C33-C51
Author(s):  
Lishan Fang
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Thin Plate ◽  
Thin Plate Spline ◽  
Adaptive Refinement ◽  
Fine Grid ◽  
Computational Costs ◽  
Uniform Grids ◽  
Error Indicators ◽  
Element Method

The discrete thin plate spline is a data fitting and smoothing technique for large datasets. Current research only uses uniform grids for this discrete smoother, which may require a fine grid to achieve a certain accuracy. This leads to a large system of equations and high computational costs. Adaptive refinement adapts the precision of the solution to reduce computational costs by refining only in sensitive regions. The error indicator is an essential part of the adaptive refinement as it identifies whether certain regions should be refined. Error indicators are well researched in the finite element method, but they might not work for the discrete smoother as data may be perturbed by noise and not uniformly distributed. Two error indicators are presented: one computes errors by solving an auxiliary problem and the other uses the bounds of the finite element error. Their performances are evaluated and compared with 2D model problems. References H. Chui and A. Rangarajan. A new point matching algorithm for non-rigid registration. Comput. Vis. Image Und., 89 (23): 114141, 2003. doi:10.1016/S1077-3142(03)00009-2. W. F. Mitchell. A comparison of adaptive refinement techniques for elliptic problems. ACM T. Math. Software, 15 (4): 326347, 1989. doi:10.1145/76909.76912. S. Roberts, M. Hegland, and I. Altas. Approximation of a thin plate spline smoother using continuous piecewise polynomial functions. SIAM J. Numer. Anal., 41(1):208234, 2003. doi:10.1137/S0036142901383296. G. Sewell. Analysis of a finite element method. Springer-Verlag, 1985. doi:10.1007/978-1-4684-6331-6. R. Sprengel, K. Rohr, and H. S. Stiehl. Thin-plate spline approximation for image registration. In P. IEEE EMBS, volume 3, pages 11901191. IEEE, 1996. doi:10.1109/IEMBS.1996.652767. L. Stals. Efficient solution techniques for a finite element thin plate spline formulation. J. Sci. Comput., 63(2):374409, 2015. doi:10.1007/s10915-014-9898-x. G. Wahba. Spline models for observational data, volume 59 of CBMS-NSF Regional Conference Series in Applied Mathematics. SIAM, 1990. doi:10.1137/1.9781611970128.

An Immersed Finite Element Method for the Elasticity Problems with Displacement Jump

2017 ◽  
Vol 9 (2) ◽  
pp. 407-428 ◽  
Author(s):  
Daehyeon Kyeong ◽  
Do Young Kwak
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Normal Stress ◽  
Basis Functions ◽  
Displacement Discontinuity ◽  
Immersed Finite Element Method ◽  
Immersed Finite Element ◽  
Uniform Grids ◽  
Elasticity Problems ◽  
Element Method

AbstractIn this paper, we propose a finite element method for the elasticity problems which have displacement discontinuity along the material interface using uniform grids. We modify the immersed finite element method introduced recently for the computation of interface problems having homogeneous jumps [20, 22]. Since the interface is allowed to cut through the element, we modify the standard Crouzeix-Raviart basis functions so that along the interface, the normal stress is continuous and the jump of the displacement vector is proportional to the normal stress. We construct the broken piecewise linear basis functions which are uniquely determined by these conditions. The unknowns are only associated with the edges of element, except the intersection points. Thus our scheme has fewer degrees of freedom than most of the XFEM type of methods in the existing literature [1,8,13]. Finally, we present numerical results which show optimal orders of convergence rates.

scholarly journals A BASIC CONVERGENCE RESULT FOR CONFORMING ADAPTIVE FINITE ELEMENTS

2008 ◽  
Vol 18 (05) ◽  
pp. 707-737 ◽  
Author(s):  
PEDRO MORIN ◽  
KUNIBERT G. SIEBERT ◽  
ANDREAS VEESER
Keyword(s):  
Finite Element ◽  
Finite Elements ◽  
Adaptive Algorithm ◽  
Convergence Result ◽  
Necessary Condition ◽  
Error Estimator ◽  
Linear Boundary ◽  
Adaptive Finite Elements ◽  
Uniform Grids ◽  
Point Type

We consider the approximate solution with adaptive finite elements of a class of linear boundary value problems, which includes problems of "saddle point" type. For the adaptive algorithm we assume the following framework: refinement relies on unique quasi-regular element subdivisions and generates locally quasi-uniform grids, the finite element spaces are conforming, nested, and satisfy the inf–sup conditions, the error estimator is reliable as well as locally and discretely efficient, and marked elements are subdivided at least once. Under these assumptions, we give a sufficient and essentially necessary condition on marking for the convergence of the finite element solutions to the exact one. This condition is not only satisfied by Dörfler's strategy, but also by the maximum strategy and the equidistribution strategy.

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