scholarly journals Nonlinearly Preconditioned FETI Solver for Substructured Formulations of Nonlinear Problems

Mathematics ◽  
2021 ◽  
Vol 9 (24) ◽  
pp. 3165
Author(s):  
Camille Negrello ◽  
Pierre Gosselet ◽  
Christian Rey
Keyword(s):  
Finite Element ◽  
Time Integration ◽  
Nonlinear Problems ◽  
Finite Element Approximation ◽  
Diffusion Problem ◽  
Implicit Time ◽  
Element Approximation ◽  
Dual Approach ◽  
Nonlinear Version ◽  
Exact Shape

We consider the finite element approximation of the solution to elliptic partial differential equations such as the ones encountered in (quasi)-static mechanics, in transient mechanics with implicit time integration, or in thermal diffusion. We propose a new nonlinear version of preconditioning, dedicated to nonlinear substructured and condensed formulations with dual approach, i.e., nonlinear analogues to the Finite Element Tearing and Interconnecting (FETI) solver. By increasing the importance of local nonlinear operations, this new technique reduces communications between processors throughout the parallel solving process. Moreover, the tangent systems produced at each step still have the exact shape of classically preconditioned linear FETI problems, which makes the tractability of the implementation barely modified. The efficiency of this new preconditioner is illustrated on two academic test cases, namely a water diffusion problem and a nonlinear thermal behavior.

GRADIENT RECOVERY FOR SINGULARLY PERTURBED BOUNDARY VALUE PROBLEMS II: TWO-DIMENSIONAL CONVECTION–DIFFUSION

2001 ◽  
Vol 11 (07) ◽  
pp. 1169-1179 ◽  
Author(s):  
HANS-GÖRG ROOS ◽  
TORSTEN LINß
Keyword(s):  
Finite Element ◽  
Finite Element Approximation ◽  
Perturbation Parameter ◽  
Diffusion Problem ◽  
Singularly Perturbed ◽  
Element Approximation ◽  
Gradient Recovery ◽  
Galerkin Finite Element Method ◽  
Galerkin Finite Element ◽  
Convection Diffusion

We consider a Galerkin finite element method that uses bilinear elements on a class of Shishkin-type meshes for a model singularly perturbed convection–diffusion problem. The recovered gradient is itself piecewise bilinear, with values at the nodes obtained by first interpolating the gradient of the finite element approximation at the centroids of the elements sharing the node. We prove a superconvergence estimate for the recovered gradient uniformly with respect to the singular perturbation parameter. Numerical experiments support our results.

An Angular Finite Element Method for the Solution of the Even-Parity Equation

2009 ◽  
Author(s):  
R. Becker ◽  
R. Koch ◽  
M. F. Modest ◽  
H.-J. Bauer
Keyword(s):  
Finite Element ◽  
Finite Element Approximation ◽  
Basis Functions ◽  
New Method ◽  
Discrete Ordinates ◽  
Test Case ◽  
Element Approximation ◽  
Promising Alternative ◽  
Element Discretization ◽  
Spatial Coordinates

The present article introduces a new method to solve the radiative transfer equation (RTE). First, a finite element discretization of the solid angle dependence is derived, wherein the coefficients of the finite element approximation are functions of the spatial coordinates. The angular basis functions are defined according to finite element principles on subdivisions of the octahedron. In a second step, these spatially dependent coefficients are discretized by spatial finite elements. This approach is very attractive, since it provides a concise derivation for approximations of the angular dependence with an arbitrary number of angular nodes. In addition, the usage of high-order angular basis functions is straightforward. In the current paper the governing equations are first derived independently of the actual angular approximation. Then, the design principles for the angular mesh are discussed and the parameterization of the piecewise angular basis functions is derived. In the following, the method is applied to two-dimensional test cases which are commonly used for the validation of approximation methods of the RTE. The results reveal that the proposed method is a promising alternative to the well-established practices like the Discrete Ordinates Method (DOM) and provides highly accurate approximations. A test case known to exhibit the ray effect in the DOM verifies the ability of the new method to avoid ray effects.

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