Anisotropic Elliptic Problems Research Articles

Block Preconditioning Methods for Asymptotic Preserving Scheme Arising in Anisotropic Elliptic Problems

Block Preconditioning Methods for Asymptotic Preserving Scheme Arising in Anisotropic Elliptic Problems

Existence and regularity of solutions of nonlinear anisotropic elliptic problem with Hardy potential

In this paper, we are interested in the existence and regularity of solutions for some anisotropic elliptic equations with Hardy potential and L m ( Ω ) data in appropriate anisotropic Sobolev spaces. The aim of this work is to get natural conditions to show the existence and regularity results for the solutions, that is related to an anisotropic Hardy inequality.

A Cell-Centered Multigrid Solver for the Finite Volume Discretization of Anisotropic Elliptic Interface Problems on Irregular Domains

A Cell-Centered Multigrid Solver for the Finite Volume Discretization of Anisotropic Elliptic Interface Problems on Irregular Domains

Solvability of nonlinear anisotropic elliptic unilateral problems with variable exponent and measure data

Solvability of nonlinear anisotropic elliptic unilateral problems with variable exponent and measure data

Nodal discrete duality numerical scheme for nonlinear diffusion problems on general meshes

Abstract Discrete duality finite volume (DDFV) schemes are known for their ability to approximate nonlinear and linear anisotropic diffusion operators on general meshes, but they possess several drawbacks. The most important drawback of DDFV is the simultaneous use of the cell and the node unknowns. We propose a discretization approach that incorporates DDFV ideas and the associated analysis techniques, but allows for a rapid elimination of the cell unknowns. Further, unlike the DDFV scheme, the new ‘Nodal Discrete Duality’ (NDD) scheme does not require specific adaptation in presence of discontinuities of the diffusion tensor along cell boundaries. We describe in detail the 2D NDD framework and its two 3D variants, focusing on the consistency properties of the discrete gradient and discrete divergence operators and on the key structural property of discrete duality. For the 2D scheme, convergence analysis is carried out and a series of numerical tests are provided for a large family of nonlinear anisotropic elliptic problems with zero Dirichlet boundary condition.

Strongly nonlinear unilateral anisotropic elliptic problem with L^1-data

Strongly nonlinear unilateral anisotropic elliptic problem with L^1-data

Solutions of an anisotropic elliptic problem involving nonlinear terms

Using variational methods, the existence and multiplicity of weak solutions for the following Neumann anisotropic problem

Existence of entropy solutions for some quasilinear anisotropic elliptic unilateral problems with variable exponents

In this paper, we shall be concerned with the study of the following quasilinear anisotropic elliptic Dirichlet problems of the type \begin{document}$\begin{equation} -\mbox{div }\;a(x,u,\nabla u) = f -\mbox{div } F \qquad \mbox{in}\quad \Omega,\quad\quad\quad\quad\quad(1) \end{equation}$ \end{document} where $ f\in L^{1}(\Omega) $ and $ F \in \prod_{i = 1}^{N} L^{p'_{i}(\cdot)}(\Omega), $ and $ a_{i}(x,u,\xi) $ are Carathéodory functions from $ \Omega \times I\!\!R \times I\!\!R^{N} $ into $ I\!\!R $, which satisfy assumptions of growth, coercivity and strict monotonicity. We prove the existence of entropy solutions for the quasilinear elliptic equation associated to the unilateral problem by relying on the penalization method, in the anisotropic variable exponent Sobolev spaces. Our approach is also based on the techniques of monotone operators in Banach spaces, the existence of weak solutions, and some approximations methods. The problems of the type $ (1) $ are very interesting from the purely mathematical point of view. On the other hand, such equations $ (1) $ appear in different contexts, in particular, the mathematical description of motions of the non-newtonien fluids; we quote for instance the electro-rheological fluids; the deformation of membrane constrained by an obstacle, the image processing and other various physical applications.

Anisotropic elliptic equations with gradient-dependent lower order terms and $ L^1 $ data

<abstract><p>We prove the existence of a weak solution for a general class of Dirichlet anisotropic elliptic problems such as $ \mathcal Au+\Phi(x, u, \nabla u) = \mathfrak{B}u+f $ in $ \Omega $, where $ \Omega $ is a bounded open subset of $ \mathbb R^N $ and $ f\in L^1(\Omega) $ is arbitrary. The principal part is a divergence-form nonlinear anisotropic operator $ \mathcal A $, the prototype of which is $ \mathcal A u = -\sum_{j = 1}^N \partial_j(|\partial_j u|^{p_j-2}\partial_j u) $ with $ p_j > 1 $ for all $ 1\leq j\leq N $ and $ \sum_{j = 1}^N (1/p_j) > 1 $. As a novelty in this paper, our lower order terms involve a new class of operators $ \mathfrak B $ such that $ \mathcal{A}-\mathfrak{B} $ is bounded, coercive and pseudo-monotone from $ W_0^{1, \overrightarrow{p}}(\Omega) $ into its dual, as well as a gradient-dependent nonlinearity $ \Phi $ with an "anisotropic natural growth" in the gradient and a good sign condition.</p></abstract>

Generalized anisotropic elliptic Wentzell problems with nonstandard growth conditions

Let Ω⊆RN be a bounded Lipschitz domain with the W1,p→(⋅)-extension property, for N≥3. We investigate the solvability and global regularity of a class of quasi-linear elliptic equations involving the anisotropic p→(⋅)-Laplace operator Δp→(⋅), with nonhomogeneous anisotropic Wentzell boundary conditions ∑i=1N|∇u|pi(⋅)−2∂u∂xiνi−Δq→(⋅),Γu+β|u|qM(⋅)−2u∋0onΓ≔∂Ω,for β∈L∞(Γ)+ with a positive lower bound, where p→∈C0,1(Ω¯)N and q→∈C0,1(Γ)N−1 fulfill 1<pm−≤pM+<∞, and 1<qm−≤qM+<∞. Here Δq→(⋅),Γ denotes the anisotropic q→(⋅)-Laplace–Beltrami operator. We first show existence and uniqueness of weak solutions for the elliptic problem, and moreover, we prove that such solutions are globally bounded over Ω¯. Key L∞-type a priori estimates for the difference of weak solutions are provided, as well as Maximum principles and inverse positivity results. At the end, we establish a sort of “nonlinear anisotropic Fredholm alternative” for the corresponding anisotropic Wentzell problem of nonstandard structure.

Existence and uniqueness result for nonlinear anisotropic elliptic unilateral problems with variable exponent and measure data

Existence and uniqueness result for nonlinear anisotropic elliptic unilateral problems with variable exponent and measure data

An exact-interface-fitted mesh generator and linearity-preserving finite volume scheme for anisotropic elliptic interface problems

Anisotropic elliptic interface problems with non-homogeneous jump conditions are of great importance but hard to solve numerically. In this work, we develop an easy-to-implement exact-interface-fitted mesh generation (EIFMG) algorithm, and then propose a linearity-preserving finite volume (FV) scheme for solving such problems. Firstly, by curving the cell edges to align with the material interface, the EIFMG algorithm produces a structured exact-interface-fitted curved quadrilateral mesh without changing the total number of cells and associated topology. Secondly, by employing the so-called linearity preserving criterion, our derived FV scheme can be exact on curved quadrilateral meshes for constant coefficient problems with a linear solution even with extremely coarse mesh, which is generally not easy for finite element or finite difference methods to achieve. Moreover, a new method for interpolating vertex unknowns with cell-centered ones is also developed for non-homogeneous jump conditions of solutions and normal fluxes. Finally, numerical results are presented to verify the second-order accuracy and linearity-preserving ability of the proposed FV scheme. As far as we know, the algorithm based on FV method in this paper seems to be the best way to handle anisotropic elliptic interface problems with nonhomogeneous jump conditions, judging from the computational performance and efficiency of many typical numerical examples.

Comparison of standard and stabilization free Virtual Elements on anisotropic elliptic problems

In this letter we compare the behaviour of standard Virtual Element Methods (VEM) and stabilization free Enlarged Enhancement Virtual Element Methods (E2V EM) with the focus on some elliptic test problems whose solution and diffusivity tensor are characterized by anisotropies. Results show that the possibility to avoid an arbitrary stabilizing part, offered by E2V EM methods, can reduce the magnitude of the error on general polygonal meshes and help convergence.

Existence of entropy solutions of the anisotropic elliptic nonlinear problem with measure data in weighted Sobolev space

This paper is devoted to study the following nonlinear anisotropic elliptic unilateral problem\begin{equation*}\begin{cases}A\,u -\mbox{div}\,\phi(u)=\mu \quad \mbox{in} \qquad \Omega \\\;u=0 \qquad \mbox{on} \quad \partial \Omega ,\end{cases}\end{equation*}where the right hand side $\,\mu\;$ belongs to $\; L^1(\Omega)+ W_{0}^{-1,\overrightarrow{p}'} (\Omega,\ \overrightarrow{\omega}^*)$. The operator $\displaystyle A\,u=-\sum_{i=1}^{N}\partial_{i}\,a_{i}(x,\ u,\ \nabla u)$ is a Leray-Lions anisotropic operator acting from $\; W_{0}^{1,\overrightarrow{p}} (\Omega,\ \overrightarrow{\omega})\;$ into its dual $\; W_{0}^{-1,\overrightarrow{p}'} (\Omega,\ \overrightarrow{\omega}^*)$ and $\phi_{i}\in C^{0}(\mathbb{R},\mathbb{R})$.

Existence and regularity results for nonlinear anisotropic unilateral elliptic problems with degenerate coercivity

Existence and regularity results for nonlinear anisotropic unilateral elliptic problems with degenerate coercivity

On some nonlinear anisotropic elliptic equations in anisotropic Orlicz space

PurposeIn the present paper, the authors will discuss the solvability of a class of nonlinear anisotropic elliptic problems (P), with the presence of a lower-order term and a non-polynomial growth which does not satisfy any sign condition which is described by an N-uplet of N-functions satisfying the Δ2-condition, within the fulfilling of anisotropic Sobolev-Orlicz space. In addition, the resulting analysis requires the development of some new aspects of the theory in this field. The source term is merely integrable.Design/methodology/approachAn approximation procedure and some priori estimates are used to solve the problem.FindingsThe authors prove the existence of entropy solutions to unilateral problem in the framework of anisotropic Sobolev-Orlicz space with bounded domain. The resulting analysis requires the development of some new aspects of the theory in this field.Originality/valueTo the best of the authors’ knowledge, this is the first paper that investigates the existence of entropy solutions to unilateral problem in the framework of anisotropic Sobolev-Orlicz space with bounded domain.

Anisotropic problem with non-local boundary conditions and measure data

We study a nonlinear anisotropic elliptic problem with non-local boundary conditions and measure data. We prove an existence and uniqueness result of entropy solution.

An FE-FD Method for Anisotropic Elliptic Interface Problems

Anisotropic elliptic interface problems are important but hard to solve either analytically or numerically. There is limited literature on numerical methods based on structured meshes. Finite eleme...

Equivalence of Entropy and Renormalized Solutions of Anisotropic Elliptic Problem in Unbounded Domains with Measure Data

Equivalence of Entropy and Renormalized Solutions of Anisotropic Elliptic Problem in Unbounded Domains with Measure Data

A new conservative finite-difference scheme for anisotropic elliptic problems in bounded domain

Highly anisotropic elliptic problems occur in many physical models that need to be solved numerically. A direction of dominant diffusion is thus introduced (called here parallel direction) along which the diffusion coefficient is several orders larger of magnitude than in the perpendicular one. In this case, finite-difference methods based on misaligned stencils are generally not designed to provide an optimal discretization, and may lead the perpendicular diffusion to be polluted by the numerical error in approximating the parallel diffusion.This paper proposes an original scheme using non-aligned Cartesian grids and interpolations aligned along a parallel diffusion direction. Here, this direction is assumed to be supported by a divergence-free vector field which never vanishes and it is supposed to be stationary in time. Based on the Support Operator Method (SOM), the self-adjointness property of the parallel diffusion operator is maintained on the discrete level. Compared with existing methods, the present formulation further guarantees the conservativity of the fluxes in both parallel and perpendicular directions. In addition, when the flow intercepts a boundary in the parallel direction, an accurate discretization of the boundary condition is presented that avoids the uncertainties of extrapolated far ghost points classically used and ensures a better accuracy of the solution. Numerical tests based on manufactured solutions show the method is able to provide accurate and stable numerical approximations in both periodic and bounded domains with a drastically reduced number of degrees of freedom with respect to non-aligned approaches.