Elliptic Boundary Value Problems Research Articles

Existence of Solutions for p(x)-Laplacian Elliptic BVPs on a Variable Sobolev Space Via Fixed Point Theorems

In this paper, we prove some existence theorems for elliptic boundary value problems within the p(x)-Laplacian on a variable Sobolev space. For this purpose, the main problem is transformed into a fixed point problem and then fixed point arguments such as Schaefer’s and Schauder’s theorems are used. Our approach involves fewer stringent assumptions on the nonlinearity function than the prior findings. An interesting example is presented to examine the validity of the theoretical findings.

Analysis of solution to an elliptic free boundary value problem equipped with a ‘bad’ data

We will study a free boundary value problem driven by a source term which is quite irregular. In the process, we will establish a monotonicity result, and the regularity of solutions to the free boundary value problem under consideration.

An Iterative Two-Grid Method for Strongly Nonlinear Elliptic Boundary Value Problems

An Iterative Two-Grid Method for Strongly Nonlinear Elliptic Boundary Value Problems

On a class of infinite semipositone problems for ( p , q ) Laplace operator

We analyze a non-linear elliptic boundary value problem that involves ( p , q ) Laplace operator, for the existence of its positive solution in an arbitrary smooth bounded domain. The non-linearity here is driven by a singular, monotonically increasing continuous function in ( 0 , ∞ ) which is eventually positive. The novelty in proving the existence of a positive solution lies in the construction of a suitable subsolution. Our contribution marks an advancement in the theory of existence of positive solutions for infinite semipositone problems in arbitrary bounded domains, whereas the prevailing theory is limited to addressing similar problems only in symmetric domains. Additionally, using the ideas pertaining to the construction of subsolution, we establish the exact behavior of the solutions of “q-sublinear” problem involving ( p , q ) Laplace operator when the parameter λ is very large. The parameter estimate that we derive is non-trivial due to the non-homogeneous nature of the operator and is of independent interest.

Eta-invariant of elliptic parameter-dependent boundary-value problems

In this paper, we study the eta-invariant of elliptic parameter-dependent boundary value problems and its main properties. Using Melrose’s approach, we de ne the eta-invariant as a regularization of the winding number of the family. In this case, the regularization of the trace requires obtaining the asymptotics of the trace of compositions of invertible parameter-dependent boundary value problems for large values of the parameter. Obtaining the asymptotics uses the apparatus of pseudodifferential boundary value problems and is based on the reduction of parameter-dependent boundary value problems to boundary value problems with no parameter.

A Hybrid High-Order Method for a Class of Strongly Nonlinear Elliptic Boundary Value Problems

A Hybrid High-Order Method for a Class of Strongly Nonlinear Elliptic Boundary Value Problems

A two-dimensional boundary value problem of elliptic type with nonlocal conjugation conditions

Abstract We consider an elliptic boundary value problem with nonlocal conjugation conditions. An a priori estimate for its weak solution in an appropriate Sobolev-like space is proved. A finite difference scheme approximating this problem is proposed and analyzed. An estimate of the convergence rate, compatible with the smoothness of the input data, is obtained.

Nine-point compact sixth-order approximation for two-dimensional nonlinear elliptic partial differential equations: Application to bi- and tri-harmonic boundary value problems

Nine point sixth order compact numerical approximations are suggested to solve 2D nonlinear elliptic partial differential equations (NLEPDEs) and for the estimation of normal derivatives on a uniform rectangular grid subject to Dirichlet boundary conditions. We deliberate error analysis and reveal that, under specific conditions, our method converges to the sixth order. In addition, we extend our technique to vector form in order to solve the system of NLEPDEs. In application, we discuss nine-point compact sixth order approximations for bi- and tri-harmonic elliptic boundary value problems. Numerical experiments are carried out on several benchmark problems including bi- and tri-harmonic equations, and verified the sixth order convergence of the proposed methods.

Simulating accurate and effective solutions of some nonlinear nonlocal two-point BVPs: Clique and QLM-clique matrix methods

This research work deals with two spectral matrix collocation algorithms based on (novel) clique functions to solve two classes of nonlinear nonlocal elliptic two-points boundary value problems (BVPs) arising in diverse physical models. The nonlinearity together with nonlocality makes the models so difficult to solve numerically. In both matrix methods by expanding the unknown solutions in terms of clique polynomials the nonlocality in the models is handled. In the first and direct technique, the clique coefficients are determined in an accurate way through solving an algebraic system of nonlinear equations. To get rid of the nonlinearity of the models and in order to gain efficacy, the quasilinearization method (QLM) is utilized in the second approach. In the latter algorithm named QLM-clique procedure, the first directly clique collocation method is applied to a family of linearized equations in an iterative manner. In particular, we show that the series expansion of clique polynomials is convergent exponentially in a weighted L2 norm. Numerous numerical simulations validate the performance of two matrix collocation schemes and demonstrate the accuracy as well as the gain in computational efficiency in terms of elapsed CPU time. The proposed matrix algorithms for computation are easy to implement and straightforward, and provide more accuracy compared to other available computational values in the literature.

Hidden convexity in the heat, linear transport, and Euler’s rigid body equations: A computational approach

A finite element based computational scheme is developed and employed to assess a duality based variational approach to the solution of the linear heat and transport PDE in one space dimension and time, and the nonlinear system of ODEs of Euler for the rotation of a rigid body about a fixed point. The formulation turns initial-(boundary) value problems into degenerate elliptic boundary value problems in (space)-time domains representing the Euler-Lagrange equations of suitably designed dual functionals in each of the above problems. We demonstrate reasonable success in approximating solutions of this range of parabolic, hyperbolic, and ODE primal problems, which includes energy dissipation as well as conservation, by a unified dual strategy lending itself to a variational formulation. The scheme naturally associates a family of dual solutions to a unique primal solution; such ‘gauge invariance’ is demonstrated in our computed solutions of the heat and transport equations, including the case of a transient dual solution corresponding to a steady primal solution of the heat equation. Primal evolution problems with causality are shown to be correctly approximated by noncausal dual problems.

Rates of robust superlinear convergence of preconditioned Krylov methods for elliptic FEM problems

AbstractThis paper considers the iterative solution of finite element discretizations of second-order elliptic boundary value problems. Mesh independent estimations are given for the rate of superlinear convergence of preconditioned Krylov methods, involving the connection between the convergence rate and the Lebesgue exponent of the data. Numerical examples demonstrate the theoretical results.

A Novel ANN-Based Radial Basis Function Collocation Method for Solving Elliptic Boundary Value Problems

Elliptic boundary value problems (BVPs) are widely used in various scientific and engineering disciplines that involve finding solutions to elliptic partial differential equations subject to certain boundary conditions. This article introduces a novel approach for solving elliptic BVPs using an artificial neural network (ANN)-based radial basis function (RBF) collocation method. In this study, the backpropagation neural network is employed, enabling learning from training data and enhancing accuracy. The training data consist of given boundary data from exact solutions and the radial distances between exterior fictitious sources and boundary points, which are used to construct RBFs, such as multiquadric and inverse multiquadric RBFs. The distinctive feature of this approach is that it avoids the discretization of the governing equation of elliptic BVPs. Consequently, the proposed ANN-based RBF collocation method offers simplicity in solving elliptic BVPs with only given boundary data and RBFs. To validate the model, it is applied to solve two- and three-dimensional elliptic BVPs. The results of the study highlight the effectiveness and efficiency of the proposed method, demonstrating its capability to deliver accurate solutions with minimal data input for solving elliptic BVPs while relying solely on given boundary data and RBFs.

Finite element study of V-shaped crack-tip fields in a three-dimensional nonlinear strain-limiting elastic body

We study the crack-tip fields in a three-dimensional (3D) plate with a V-shaped crack employing Rajagopal’s theory of elasticity with the strain-limiting effect. The classical linear elastic fracture mechanics (LEFM) theory bears a well-documented inconsistency, i.e., for a static crack problem under the traction-free boundary condition, the LEFM theory predicts singular crack-tip strains. Such results are inconsistent with the constitutive relation derived using the assumption of uniform bound for the strain values. Instead, Rajagopal’s recent theory of elasticity provides implicit constitutive relations between the linearized strain and Cauchy stress in which one can impose an a-priori upper bound for strains. We formulate the strain-limiting theory comprehensively in this article. The linearized strain is expressed as a nonlinear function of the Cauchy stress. The equation for the equilibrium of the linear momentum under a special type of nonlinear constitutive relation reduces to a second-order quasilinear elliptic boundary value problem (BVP). Using a Picard-type linearization and a continuous Galerkin-type finite element technique, we solve several BVPs with tensile and shear displacement loading. The numerical results for the strains, the stresses, the stress intensity factor (SIF), and strain energy density (SED) are obtained for different Poisson values, strain-limiting parametric values, and the V-shaped angles. The growth of the near-tip strains is far less than that of stresses, and the results for both the SIF and SED are also consistent with the results from LEFM. Our work demonstrates that the framework of Rajagopal’s theory of elasticity can provide a basis for developing physically meaningful and mathematically well-posed BVPs to study evolution of cracks, damage, nucleation, or failure in elastic bodies.

Interaction of Boundary Singular Points in an Elliptic Boundary Value Problem

The paper continues the construction of the Lp-theory of elliptic Dirichlet and Neumann boundary value problems with discontinuous piecewise constant coefficients in divergent form for an unbounded domain R2 with a piecewisesmooth noncompact Lipschitz boundary and C1 smooth discontinuity lines of the coefficients. An earlier constructed Lp-theory is generalized to the case of different smallest eigenvalues corresponding to a finite and an infinite singular point, and the effect of their interaction is further studied in the class of functions with first derivatives from Lp( ) in the entire range of the exponent p (1, )..

Interaction of Boundary Singular Points in an Elliptic Boundary Value Problem

Interaction of Boundary Singular Points in an Elliptic Boundary Value Problem

Nonlinear elliptic boundary value problems with convection term and Hardy potential

Abstract In this paper, we deal with a nonlinear elliptic problems that incorporate a Hardy potential and a nonlinear convection term. We establish the existence and regularity of solutions under various assumptions concerning the summability of the source term f.

Positive Jacobian constraints for elliptic boundary value problems with piecewise-regular coefficients arising from multi-wave inverse problems

Multi-wave inverse problems are indirect imaging methods using the interaction of two different imaging modalities. One brings spatial accuracy, and the other contrast sensitivity. The inversion method typically involve two steps. The first step is devoted to accessing internal datum of quantities related to the unknown parameters being observed. The second step involves recovering the parameters themselves from the internal data. To perform that inversion, a typical requirement is that the Jacobian of fields involved does not vanish. A number of authors have considered this problem in the past two decades, and a variety of methods have been developed. Existing techniques require Hölder continuity of the parameters to be reconstructed. In practical applications, the medium may present embedded elements, with distinct physical properties, leading to discontinuous coefficients. In this article we explain how a Jacobian constraint can be imposed in the piecewise regular case, when the physical model is a divergence form second order linear elliptic boundary value problem.

A Lagrangian method for slurry flow modeling in hydraulic fractures

We developed a new numerical simulator of slurry flow in hydraulic fractures. It is based on a continuum model developed under the assumptions of lubrication theory. We consider slurry containing fluid and proppant components. Proppant particles settle under gravity and bridge in the flow domain. Fluid fluxes are calculated from an elliptic boundary value problem for given fracture geometry and sources/sinks distribution. We solve it using a conservative finite volume method and then apply empirical closure relations to get the velocity field for solids. The transport of slurry components is described by nonlinear hyperbolic equations with fluxes depending on proppant concentration and particle size/fracture width ratio. These equations admit solutions with concentration shock waves and rarefaction fans. They must be represented accurately in the computational algorithm. We utilize an Eulerian approach for the discretization of nonlinear proppant fluxes. After getting the fluxes on the interfaces between Eulerian grid cells we follow a Lagrangian approach to simulate the proppant transport along the given velocity field. In this study we discuss key implementation aspects: time step selection, regularization, fluxes discretization, seeding, reseeding, and movement of Lagrangian particles. We verify the simulator on benchmark solutions focused on advection, batch proppant settling, slurry slumping and fluid displacement by proppant.

Discontinuous Galerkin methods for an elliptic optimal control problem with a general state equation and pointwise state constraints

We investigate discontinuous Galerkin methods for an elliptic optimal control problem with a general state equation and pointwise state constraints on general polygonal domains. We show that discontinuous Galerkin methods for general second-order elliptic boundary value problems can be used to solve the elliptic optimal control problems with pointwise state constraints. We establish concrete error estimates and numerical experiments are shown to support the theoretical results.

Approximate controllability of a semilinear elliptic problem in periodically perforated domains

In this paper, we study the controllability of a semilinear second order elliptic boundary value problem in a periodically perforated domain, where the control is distributed in an open subset of the domain. Compared to existing literature, the holes in this control problem can intersect the control zone. We study L2 and H1-approximate controllability for the problem and we prove the homogenization results for the control problem in an ε-periodically perforated domain. The holes are of same size as ε with the coefficients of state equation, and the cost functional being rapidly oscillating.