Quasilinear Elliptic Partial Differential Equations Research Articles

High precision compact numerical approximation in exponential form for the system of 2D quasilinear elliptic BVPs on a discrete irrational region

This article presents a new approximation of order four in exponential form for two-dimensional (2D) quasilinear partial differential equation (PDE) of elliptic form with solution domain being irrational. It is further extended for application to a system of quasilinear elliptic PDEs with Dirichlet boundary conditions (DBCs). The main highlights of the method framed in this article are as under:•It uses a 9-point stencil with unequal mesh to approach the solution. The error analysis is discussed to authenticate the order of convergence of the proposed numerical approximation.•Various validating problems, for instance the Burgers’ equation, Poisson equation in cylindrical coordinates, Navier-Stokes (NS) equations in rectangular and cylindrical coordinates are solved using the proposed techniques to depict their stability. The proposed approximation produces solution free of oscillations for large values of Reynolds Number in the vicinity of a singularity.•The results of the proposed method are superior in comparison to the existing methods of [49] and [56].

The virtual element method for a minimal surface problem

In this paper we consider the Virtual Element discretization of a minimal surface problem, a quasi-linear elliptic partial differential equation modeling the problem of minimizing the area of a surface subject to a prescribed boundary condition. We derive an optimal error estimate and present several numerical tests assessing the validity of the theoretical results.

High-resolution compact numerical method for the system of 2D quasi-linear elliptic boundary value problems and the solution of normal derivatives on an irrational domain with engineering applications

In this paper, we present a novel approach to attain fourth-order approximate solution of 2D quasi-linear elliptic partial differential equation on an irrational domain. In this approach, we use nine grid points with dissimilar mesh in a single compact cell. We also discuss appropriate fourth-order numerical methods for the solution of the normal derivatives on a dissimilar mesh. The method has been protracted for solving system of quasi-linear elliptic equations. The convergence analysis is discussed to authenticate the proposed numerical approximation. On engineering applications, we solve various test problems, such as linear convection–diffusion equation, Burgers’equation, Poisson equation in singular form, NS equations, bi- and tri-harmonic equations and quasi-linear elliptic equations to show the efficiency and accuracy of the proposed methods. A comprehensive comparative computational experiment shows the accuracy, reliability and credibility of the proposed computational approach.

Discrete comparison principles for quasilinear elliptic PDE

Comparison principles are developed for piecewise linear finite element approximations of quasilinear elliptic partial differential equations. We consider the analysis of a class of nonmonotone Leray-Lions problems featuring both nonlinear solution and gradient dependence in the principal coefficient, and a solution dependent lower-order term. Sufficient local and global conditions on the discretization are found for conforming finite element solutions to satisfy a comparison principle, which implies uniqueness of the solution. For problems without a lower-order term, our analysis shows the meshsize is only required to be locally controlled, based on the variance of the computed solution over each element. We include a discussion of the simpler semilinear case where a linear algebra argument allows a sharper mesh condition for the lower order term.

Bang-Bang and Multiple Valued Optimal Solutions of Control Problems Related to Quasi-linear Elliptic Equations

This paper investigates two optimal control problems in which the state equation is a quasi-linear elliptic partial differential equation. In the first problem the admissible set comprises functions with range in the interval [0,1] with prescribed integral. We show the optimal solution is unique and it is of bang-bang type. In the second problem the admissible set is a particular class of simple functions. We will show again the optimal solution is unique and derive the corresponding optimality condition.

A cubic B-spline semi-analytical algorithm for simulation of 3D steady-state convection-diffusion-reaction problems

In this work, a new cubic B-spline-based semi-analytical algorithm is presented for solving 3D anisotropic convection-diffusion-reaction (CDR) problems in the inhomogeneous medium. The mathematical model is expressed by the quasi-linear second-order elliptic partial differential equations (EPDE) with mixed derivatives and variable coefficients. The final approximation is obtained as a sum of the rough primary solution and the modified spline interpolants with free parameters. The primary solution mathematically satisfies boundary conditions. Thus, the free parameters of interpolants are chosen to satisfy the governing equation in the solution domain. The numerical examples demonstrate the high accuracy of the proposed method in solving 3D CDR problems in single- and multi-connected domains.

Forward-backward stochastic differential equations on infinite horizon and quasilinear elliptic PDEs

A class of infinite horizon forward-backward stochastic differential equations (FBSDEs) is investigated. Under some monotonicity conditions, the existence and uniqueness of solutions in an arbitrarily large space for FBSDEs on infinite horizon is obtained. The probabilistic interpretations for a large class of quasilinear elliptic partial differential equations (PDEs) in a global space is then given by virtue of the solutions of FBSDEs on infinite horizon.

Regularity of solutions of quasi-linear elliptic equations with [formula omitted] coefficients

Let D be an bounded region in Rn. The regularity of solutions of a family of quasilinear elliptic partial differential equations is studied, one example being Δnu=Vun−1. The coefficients are assumed to be in the space Llogm⁡L(D) for m>n−1. Using a Moser iteration argument coupled with the Moser-Trudinger inequality, a local L∞ bound on the solution u is proven. A Harnack-type inequality is then proven. These results are shown to be sharp with respect to m. Then essential continuity of u is proven, and away from the boundary a bound on the modulus of continuity.

Two‐Grid hp‐DGFEMs on Agglomerated Coarse Meshes

AbstractWe generalise the a priori error analysis of two‐grid hp‐version discontinuous Galerkin finite element methods for strongly monotone second‐order quasilinear elliptic partial differential equations to the case when coarse meshes consisting of general agglomerated polytopic elements are employed.

Reconstruction for the coefficients of a quasilinear elliptic partial differential equation

In this paper we consider an inverse coefficients problem for a quasilinear elliptic equation of divergence form ∇⋅C→(x,∇u(x))=0, in a bounded smooth domain Ω. We assume that C⃗(x,p→)=γ(x)p→+b→(x)|p→|2+O(|p→|3), by expanding C⃗(x,p→) around p→=0. We give a reconstruction method for γ and b→ from the Dirichlet to Neumann map defined on ∂Ω.

Compact half step approximation in exponential form for the system of 2D second-order quasi-linear elliptic partial differential equations

ABSTRACTThis paper deals with devising a novel exponential scheme, implicit in nature, using half step discretization of order four for computing the numerical solution of quasi-linear elliptic partial differential equations of the type , . Nine mesh points in a single compact cell are being used in this method. An elaborate convergence analysis of the proposed method has been worked out. The technique for a scalar equation is eventually extended to solve the systems of quasi-linear elliptic equations. The technique is then applied to noteworthy problems and computational outcomes corroborate the theoretical rate of convergence. Linear and non-linear singular problems are tackled separately ensuring the usage of nine spatial grid points of a single computing cell.

A matrix analysis approach to discrete comparison principles for nonmonotone PDE

We present a linear algebra approach to establishing a discrete comparison principle for a nonmonotone class of quasilinear elliptic partial differential equations. In the absence of a lower order term, local conditions on the mesh are required to establish the comparison principle and uniqueness of the piecewise linear finite element solution. We consider the assembled matrix corresponding to the linearized problem satisfied by the difference of two solutions to the nonlinear problem. Monotonicity of the assembled matrix establishes a maximum principle for the linear problem and a comparison principle for the nonlinear problem. The matrix analysis approach to the discrete comparison principle yields sharper constants and more relaxed mesh conditions than does the argument by contradiction used in previous work.

Operator compact exponential approximation for the solution of the system of 2D second order quasilinear elliptic partial differential equations

In this paper, we suggest a new exponential implicit method based on full step discretization of order four for the solution of quasilinear elliptic partial differential equation of the form A ( x,y,z ) z_{xx} +C ( x,y,z ) z_{yy} =k ( x,y,z, z_{x}, z_{y} ), 0< x,y<1. In this method a single compact cell consisting of nine nodal points is used. Convergence analysis of the said method is discussed in detail. The developed method is successfully applied to solving problems in polar coordinates. The method for scalar equation is eventually applied to solving the system of quasilinear elliptic equations. To measure the rationality and precision, the method is applied to solving several noteworthy problems and numerical results are provided to exhibit the effectiveness of the method.

Cloaking for a quasi-linear elliptic partial differential equation

In this article we consider cloaking for a quasi-linear elliptic partial differential equation of divergence type defined on a bounded domain in $\mathbb{R}^N$ for $N=2,3$. We show that a perfect cloak can be obtained via a singular change of variables scheme and an approximate cloak can be achieved via a regular change of variables scheme. These approximate cloaks though non-degenerate are anisotropic. We also show, within the framework of homogenization, that it is possible to get isotropic regular approximate cloaks. This work generalizes to quasi-linear settings previous work on cloaking in the context of Electrical Impedance Tomography for the conductivity equation.

Solutions to p(x)-Laplace type equations via nonvariational techniques

In this article, we consider a class of nonlinear Dirichlet problems driven by a Leray-Lions type operator with variable exponent. The main result establishes an existence property by means of nonvariational arguments, that is, nonlinear monotone operator theory and approximation method. Under some natural conditions, we show that a weak limit of approximate solutions is a solution of the given quasilinear elliptic partial differential equation involving variable exponent.

Quasi-linear PDEs and forward–backward stochastic differential equations: Weak solutions

In this paper, we study the existence, uniqueness and the probabilistic representation of the weak solutions of quasi-linear parabolic and elliptic partial differential equations (PDEs) in the Sobolev space Hρ1(Rd). For this, we study first the solutions of forward–backward stochastic differential equations (FBSDEs) with smooth coefficients, regularity of solutions and their connection with classical solutions of quasi-linear parabolic PDEs. Then using the approximation procedure, we establish their convergence in the Sobolev space to the solutions of the FBSDES in the space Lρ2(Rd;Rd)⊗Lρ2(Rd;Rk)⊗Lρ2(Rd;Rk×d). This gives a connection with the weak solutions of quasi-linear parabolic PDEs. Finally, we study the unique weak solutions of quasi-linear elliptic PDEs using the solutions of the FBSDEs on infinite horizon.

A note on the steady flow of Newtonian fluids with pressure dependent viscosity in a rectangular duct

Unidirectional steady flow of Newtonian liquids with a pressure-dependent viscosity in a rectangular duct is considered. Governing momentum equation is reduced to a quasilinear second order elliptic partial differential equation. We give an analytical solution to the governing equation, and investigate the effect of aspect ratio and pressure coefficient on the velocity profiles numerically.

A New Compact Off-Step Discretization for the System of 2D Quasi-Linear Elliptic Equations on Unequal Mesh

We propose a new compact finite-difference discretization of O(k 2 + k 2 h 2 + h 4) using unequal mesh sizes h > 0 and k > 0 in x- and y- coordinate directions, respectively, for the solution of the system of two-dimensional quasi-linear elliptic partial differential equations subject to the appropriate Dirichlet boundary conditions. We use only three function evaluations in comparison to the five function O(k 2 + k 2 h 2 + h 4) convergence of the proposed finite difference scheme. Some numerical examples are given to illustrate the effectiveness of the proposed methods.

Simultaneous identification of diffusion and absorption coefficients in a quasilinear elliptic problem

In this work, we consider the identifiability of two coefficients a(u) and c(x) in a quasilinear elliptic partial differential equation from the observation of the Dirichlet-to-Neumann map. We use a linearization procedure due to Isakov (1993 Arch. Ration. Mech. Anal. 124 1–12) and special singular solutions to first determine a(0) and c(x) for x ∈ Ω. Based on this partial result, we are then able to determine a(u) for by an adjoint approach.

A new compact high order off-step discretization for the system of 2D quasi-linear elliptic partial differential equations

A new fourth-order difference method for solving the system of two-dimensional quasi-linear elliptic equations is proposed. The difference scheme referred to as off-step discretization is applicable directly to the singular problems and problems in polar coordinates. Also, new fourth-order methods for obtaining the first-order normal derivatives of the solution are developed. The convergence analysis of the proposed method is discussed in details. The methods are applied to many physical problems to illustrate their accuracy and efficiency. MSC: 65N06.