scholarly journals Convergence of nonlinear numerical approximations for an elliptic linear problem with irregular data

2021 ◽  
Author(s):  
Robert EYMARD ◽  
David Maltese
Keyword(s):  
Finite Element ◽  
Weak Solution ◽  
Control Volume ◽  
Weak Sense ◽  
Finite Element Approximation ◽  
Measure Data ◽  
Regularity Conditions ◽  
Element Approximation ◽  
Upstream Weighting ◽  
The Right

This work is devoted to the study of the approximation, using two nonlinear numerical methods, of a  linear elliptic problem with measure data and heterogeneous anisotropic diffusion matrix. Both methods  show convergence properties to a continuous solution of the problem in a weak sense, through the change  of variable u = ψ(v), where ψ is a well chosen diffeomorphism between (−1, 1) and R, and v is valued  in (−1, 1). We first study a nonlinear finite element approximation on any simplicial grid. We prove the existence of a discrete solution, and, under standard regularity conditions, we prove its convergence to a  weak solution of the problem by applying Hölder and Sobolev inequalities. Some numerical results, in 2D  and 3D cases where the solution does not belong to H 1(Ω), show that this method can provide accurate  results. We then construct a numerical scheme which presents a convergence property to the entropy  weak solution of the problem in the case where the right-hand side belongs to L1 . This is achieved owing  to a nonlinear control volume finite element (CVFE) method, keeping the same nonlinear reformulation,  and adding an upstream weighting evaluation and a nonlinear p−Laplace vanishing stabilisation term.

scholarly journals Finite element approximation of steady flows of colloidal solutions

2021 ◽  
Author(s):  
Andrea Bonito ◽  
Vivette Girault ◽  
Diane Guignard ◽  
Kumbakonam R. Rajagopal ◽  
Endre Süli
Keyword(s):  
Finite Element ◽  
Weak Solution ◽  
Partial Differential Equations ◽  
Differential Equations ◽  
Nonlinear Partial Differential Equations ◽  
Finite Element Approximation ◽  
Element Approximation ◽  
Cauchy Stress ◽  
Partial Differential ◽  
Fixed Point Algorithm

We consider the mathematical analysis and numerical approximation of a system of nonlinear partial differential equations that arises in models that have relevance to  steady isochoric flows of colloidal suspensions. The symmetric velocity gradient is assumed to be a monotone nonlinear function of the deviatoric part of the Cauchy stress tensor. We prove the existence of a weak solution to the problem, and under the additional assumption that the nonlinearity involved in the constitutive relation is Lipschitz continuous we also prove uniqueness of the weak solution. We then construct mixed finite element approximations of the system using both conforming and nonconforming finite element spaces. For both of these we prove the convergence of the method to the unique weak solution of the problem, and in the case of the conforming method we provide a bound on the error between the analytical solution and its finite element approximation in terms of the best approximation error from the finite element spaces. We propose first a Lions-Mercier type iterative method and next a classical fixed-point algorithm to solve the finite-dimensional problems resulting from the finite element discretisation of the system of nonlinear partial differential equations under consideration and present numerical experiments that illustrate the practical performance of the proposed numerical method.

scholarly journals Fully discrete finite element approximation of unsteady flows of implicitly constituted incompressible fluids

2019 ◽  
Vol 40 (2) ◽  
pp. 801-849 ◽  
Author(s):  
Endre Süli ◽  
Tabea Tscherpel
Keyword(s):  
Finite Element ◽  
Weak Solution ◽  
Unsteady Flows ◽  
Finite Element Approximation ◽  
Approximate Solutions ◽  
Element Approximation ◽  
Cauchy Stress ◽  
Fully Discrete ◽  
Backward Euler ◽  
Implicit Constitutive Theory

Abstract Implicit constitutive theory provides a very general framework for fluid flow models, including both Newtonian and generalized Newtonian fluids, where the Cauchy stress tensor and the rate of strain tensor are assumed to be related by an implicit relation associated with a maximal monotone graph. For incompressible unsteady flows of such fluids, subject to a homogeneous Dirichlet boundary condition on a Lipschitz polytopal domain $\varOmega \subset \mathbb{R}^d$, $d \in \{2,3\}$, we investigate a fully discrete approximation scheme, using a spatial mixed finite element approximation on general shape-regular simplicial meshes combined with backward Euler time-stepping. We consider the case when the velocity field belongs to the space of solenoidal functions contained in $\textrm{L}^\infty (0,T;\textrm{L}^2(\varOmega )^d)\cap \textrm{L}^q(0,T;\textrm{W}^{1,q}_0(\varOmega )^d)$ with $q\in \left (2d/(d+2), \infty \right )$, which is the maximal range of $q$ with respect to existence of weak solutions. In order to facilitate passage to the limit with the discretization parameters for the sub-range $q \in \left (2d/(d+2), (3d+2)/(d+2) \right )$, we introduce a regularization of the momentum equation by means of a penalty term, and first show convergence of a subsequence of approximate solutions to a weak solution of the regularized problem; we then pass to the limit with the regularization parameter. This is achieved by the use of a solenoidal parabolic Lipschitz truncation method, a local Minty-type monotonicity result, and various weak compactness techniques. For $q \geq (3d+2)/(d+2)$ convergence of a subsequence of approximate solutions to a weak solution can be shown directly, without the regularization term.

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.

Finite Element Approximation of the p-Laplacian

1993 ◽  
Vol 61 (204) ◽  
pp. 523 ◽  
Author(s):  
John W. Barrett ◽  
W. B. Liu
Keyword(s):  
Finite Element ◽  
Finite Element Approximation ◽  
Element Approximation
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