BDDC Preconditioners for Spectral Element Discretizations of Almost Incompressible Elasticity in Three Dimensions

2010 ◽  
Vol 32 (6) ◽  
pp. 3604-3626 ◽  
Author(s):  
Luca F. Pavarino ◽  
Olof B. Widlund ◽  
Stefano Zampini
Keyword(s):  
Spectral Element ◽  
Three Dimensions ◽  
Incompressible Elasticity ◽  
Element Discretizations

Preconditioning Methods for the Penalty Function FEM Discretizations of 3D Nearly Incompressible Elasticity Problems

2016 ◽  
Vol 13 (05) ◽  
pp. 1650029
Author(s):  
Yingxiong Xiao ◽  
Zhenyou Li ◽  
Lei Zhou
Keyword(s):  
Penalty Function ◽  
Mixed Boundary ◽  
Fem Analysis ◽  
Three Dimensions ◽  
Preconditioned Conjugate Gradient ◽  
System Of Equations ◽  
Elasticity Problems ◽  
Overall Efficiency ◽  
Incompressible Elasticity ◽  
Nearly Incompressible Elasticity

Penalty function finite element method (FEM) is an efficient method for the solution of the nearly incompressible elasticity problems in three dimensions. In order to improve the overall efficiency, some faster solvers for the corresponding FEM system of equations must be presented. Since the resulting system of equations is symmetric positive definite and highly ill-conditioned, the preconditioned conjugate gradient (PCG) method is one of the most efficient methods for solving such FEM equations. In this paper, we have first presented some types of efficient PCG methods including [Formula: see text]-PCG and [Formula: see text]-PCG with two different block diagonal inverses as a preconditioner, respectively, and including the RS-GAMG-PCG based on the global matrix for the penalty function FEM equations of three-dimensional (3D) nearly incompressible elasticity problems with mixed boundary conditions. Furthermore, we apply these methods to the solution of the penalty function quadratic discretizations for the cantilever and the Cook’s membrane problems. The numerical results have shown that much more efficient PCG methods for 3D nearly incompressible problems can be obtained by using the known information that is easily available in most FEM applications, for instance, the type of the partial differential equations (PDEs) considered and the number of physical unknowns residing in each grid, and by combining the coarsening techniques used in the classic algebraic multigrid (AMG) method. This will greatly improve the overall efficiency of the penalty function FEM analysis.

scholarly journals Finite Element Preconditioning on Spectral Element Discretizations for Coupled Elliptic Equations

2012 ◽  
Vol 2012 ◽  
pp. 1-16 ◽  
Author(s):  
JongKyum Kwon ◽  
Soorok Ryu ◽  
Philsu Kim ◽  
Sang Dong Kim
Keyword(s):  
Optimal Control ◽  
Finite Element ◽  
Optimal Control Problem ◽  
Control Problem ◽  
Elliptic System ◽  
Elliptic Equations ◽  
Spectral Element ◽  
Uniform Bounds ◽  
Element System ◽  
Element Discretizations

The uniform bounds on eigenvalues ofB^h2−1A^N2are shown both analytically and numerically by theP1finite element preconditionerB^h2−1for the Legendre spectral element systemA^N2u¯=f¯which is arisen from a coupled elliptic system occurred by an optimal control problem. The finite element preconditioner is corresponding to a leading part of the coupled elliptic system.

In-depth comparison of the spectral-element and finite-element method for 3D CSEM forward modelling

2021 ◽  
Author(s):  
Michael Weiss ◽  
Paula Rulff ◽  
Thomas Kalscheuer
Keyword(s):  
Finite Element ◽  
Spectral Element ◽  
Three Dimensions ◽  
The European Union ◽  
Forward Modelling ◽  
Weighted Residual Method ◽  
Horizon 2020 ◽  
Advantages And Disadvantages ◽  
Hexahedral Elements ◽  
Element Method

<p>We developed two forward modelling approaches to simulate 3-dimensional land-based controlled-source electromagnetic (CSEM) problems in frequency domain with hexahedral spectral-element meshes and tetrahedral finite-element meshes. In recent years, the geo-electromagnetic community made a lot of progress in modelling and inversion of EM data in three dimensions using a variety of approaches. The available software is used to verify the accuracy of newly developed codes, which apply e.g. different element shapes or interpolation schemes. However, a direct comparison in terms of advantages and disadvantages of different modelling strategies, especially discretisation methods in 3D, is often not focused on in publications.</p><p>Having two modelling codes and their developers available at the same place, gives us the unique opportunity to compare the approaches in a very detailed way. Our spectral-element as well as our finite-element solution is based on Galerkin’s weighted residual method and we solve the electromagnetic diffusion equations for the total electric field on the element edges.<br>The main differences between both codes are the choice and order of the interpolation functions and the discretisation of the modelling domain employing hexahedral and tetrahedral elements. While the tetrahedral meshes used in our finite-element approach are known for being able to properly resolve complex structures in the subsurface, this issue is addressed in the spectral-element method by utilising curvilinear instead of orthogonal hexahedral elements.</p><p>In this contribution, we focus on the comparison of both approaches for a simple 1D model and a complex 3D model in terms of accuracy, effort in mesh generation and computational resources such as simulation time and memory requirement. Moreover, we contrast the influence of mesh discretisation on the solution for the two methods as well as the order of approximation. A preliminary test simulation of a  model consisting of a conductive body buried within a resistive background covered by a thin conductive layer yielded comparable results in terms of accuracy. It also revealed significant differences concerning the mesh discretisation meaning the solution's dependency on the meshing of the model domain.<br><br>Acknowledgements: This work was partly funded by Uppsala’s Center for Interdisciplinary<br>Mathematics and the Smart Exploration project, which has received funding from the European Union’s Horizon 2020 research and innovation programme under grant agreement No.775971.</p>

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