Nonlinear Elliptic Partial Differential Equations Research Articles

Concavity principles for nonautonomous elliptic equations and applications

In the study of concavity properties of positive solutions to nonlinear elliptic partial differential equations the diffusion and the nonlinearity are typically independent of the space variable. In this paper we obtain new results aiming to get almost concavity results for a relevant class of anisotropic semilinear elliptic problems with spatially dependent source and diffusion.

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.

Efficient and accurate nonlinear model reduction via first-order empirical interpolation

We present a model reduction approach that extends the original empirical interpolation method to enable accurate and efficient reduced basis approximation of parametrized nonlinear partial differential equations (PDEs). In the presence of nonlinearity, the Galerkin reduced basis approximation remains computationally expensive due to the high complexity of evaluating the nonlinear terms, which depends on the dimension of the truth approximation. The empirical interpolation method (EIM) was proposed as a nonlinear model reduction technique to render the complexity of evaluating the nonlinear terms independent of the dimension of the truth approximation. We introduce a first-order empirical interpolation method (FOEIM) that makes use of the partial derivative information to construct an inexpensive and stable interpolation of the nonlinear terms. We propose two different FOEIM algorithms to generate interpolation points and basis functions. We apply the FOEIM to nonlinear elliptic PDEs and compare it to the Galerkin reduced basis approximation and the EIM. Numerical results are presented to demonstrate the performance of the three reduced basis approaches.

A deep neural network approach for parameterized PDEs and Bayesian inverse problems

We consider the simulation of Bayesian statistical inverse problems governed by large-scale linear and nonlinear partial differential equations (PDEs). Markov chain Monte Carlo (MCMC) algorithms are standard techniques to solve such problems. However, MCMC techniques are computationally challenging as they require a prohibitive number of forward PDE solves. The goal of this paper is to introduce a fractional deep neural network (fDNN) based approach for the forward solves within an MCMC routine. Moreover, we discuss some approximation error estimates. We illustrate the efficiency of fDNN on inverse problems governed by nonlinear elliptic PDEs and the unsteady Navier–Stokes equations. In the former case, two examples are discussed, respectively depending on two and 100 parameters, with significant observed savings. The unsteady Navier–Stokes example illustrates that fDNN can outperform existing DNNs, doing a better job of capturing essential features such as vortex shedding.

Localized Model Reduction for Nonlinear Elliptic Partial Differential Equations: Localized Training, Partition of Unity, and Adaptive Enrichment

Localized Model Reduction for Nonlinear Elliptic Partial Differential Equations: Localized Training, Partition of Unity, and Adaptive Enrichment

The Neumann Problem for a Class of Fully Nonlinear Elliptic Partial Differential Equations

In this paper, we establish a global $C^2$ estimates to the Neumann problem for a class of fullly nonlinear elliptic equations. By the method of continuity, we establish the existence theorem of $k$-admissible solutions of the Neumann problems.

Gradient method with AFEM for parameter-estimation

We consider the adaptive finite element discretization of parameter estimation problems for nonlinear elliptic partial differential equations. The idea is to use a gradient method on the finite-dimensional parameter space for the minimization of the least-squares residual. Since the gradient involves solution of partial differential equations, it is not accesable, and is replaced by an approximation obtained by finite elements. This results into a perturbed gradient method. We use an (a posteriori) error estimator to control the accuracy of the gradient approximation and propose an algorithm, which links the estimator to the progress of the iteration. We show convergence of the algorithm under typical structural assumptions.

Solving Nonlinear Elliptic PDEs in 2D and 3D Using Polyharmonic Splines and Low-Degree of Polynomials

In this paper, the improved localized method of approximated particular solutions (ILMAPS) using polyharmonic splines (PHS) together with a low-degree of polynomial basis is used to approximate solutions of various nonlinear elliptic Partial Differential Equations (PDEs). The method is completely meshfree, and it uses a radial basis function (RBF) that has no shape parameters. The discretization process is done through a simple collocation technique on a set of points in the local domain of influence. Resulted system of nonlinear algebraic equations is solved by the Picard method.The performance of the proposed method is tested on various nonlinear elliptical problems, including the Poisson-type problems in 2D and 3D with constant or variable coefficients on rectangular or irregular domains and the Poisson–Boltzmann equation with Dirichlet boundary conditions or mixed boundary conditions. The effect of domain shapes in 2D and 3D, types of boundary conditions, and degrees of PHS, and order of polynomial basis are examined. The performance of the method is compared with other bases such as multiquadrics (MQ) basis functions, and with results reported in the literature (method of particular solutions using polynomials). The numerical experiments suggest that ILMAPS with polyharmonic splines yields considerably superior accuracy than other RBFs as well as other approaches reported in the literature for solving nonlinear elliptic PDEs.

Global regularity for a class of fully nonlinear PDEs with unbalanced variable degeneracy

AbstractWe establish the existence and sharp global regularity results (, and estimates) for a class of fully nonlinear elliptic Partial Differential Equations (PDEs) with unbalanced variable degeneracy. In a precise way, the degeneracy law of the model switches between two different kinds of degenerate elliptic operators of variable order, according to the null set of a modulating function . The model case in question is given by for a bounded, regular, and open set , and appropriate continuous data , , and . Such sharp regularity estimates generalize and improve, to some extent, earlier ones via geometric treatments. Our results are consequences of geometric tangential methods and make use of compactness, localized oscillating, and scaling techniques. In the end, our findings are applied in the study of a wide class of nonlinear models.

Normalized Wolfe-Powell-type local minimax method for finding multiple unstable solutions of nonlinear elliptic PDEs

The local minimax method (LMM) proposed by Li and Zhou (2001) and Li and Zhou (2002) is an efficient method to solve nonlinear elliptic partial differential equations (PDEs) with certain variational structures for multiple solutions. The steepest descent direction and the Armijo-type step-size search rules are adopted in Li and Zhou (2002) and play a significant role in the performance and convergence analysis of traditional LMMs. In this paper, a new algorithm framework of the LMMs is established based on general descent directions and two normalized (strong) Wolfe-Powell-type step-size search rules. The corresponding algorithm framework named as the normalized Wolfe-Powell-type LMM (NWP-LMM) is introduced with its feasibility and global convergence rigorously justified for general descent directions. As a special case, the global convergence of the NWP-LMM algorithm combined with the preconditioned steepest descent (PSD) directions is also verified. Consequently, it extends the framework of traditional LMMs. In addition, conjugate gradient-type (CG-type) descent directions are utilized to speed up the NWP-LMM algorithm. Finally, extensive numerical results for several semilinear elliptic PDEs are reported to profile their multiple unstable solutions and compared for different algorithms in the LMM’s family to indicate the effectiveness and robustness of our algorithms. In practice, the NWP-LMM combined with the CG-type direction indeed performs much better than its known LMM companions.

On the Regularity of Non-scattering Anisotropic Inhomogeneities

In this paper we examine the necessary conditions for an anisotropic inhomogeneous medium to be non-scattering at a single wave number and for a single incident field. These conditions are expressed in terms of the regularity of the boundary of the inhomogeneity. We assume that the coefficients, characterizing the constitutive material properties of the medium, are sufficiently smooth, and that the incident wave is appropriately non-degenerate. Our analysis utilizes the Hodograph transform as well as regularity results for nonlinear elliptic partial differential equations. Our approach requires that the boundary a-priori is of class $$C^{1,\alpha }$$ for some $$0<\alpha <1$$ .

Differential geometric global smoothings of simple normal crossing complex surfaces with trivial canonical bundle

AbstractLetXXbe a simple normal crossing (SNC) compact complex surface with trivial canonical bundle which includes triple intersections. We prove that ifXXisdd-semistable, then there exists a family of smoothings in a differential geometric sense. This can be interpreted as a differential geometric analogue of the smoothability results due to Friedman, Kawamata-Namikawa, Felten-Filip-Ruddat, Chan-Leung-Ma, and others in algebraic geometry. The proof is based on an explicit construction of local smoothings around the singular locus ofXX, and the first author’s existence result of holomorphic volume forms on global smoothings ofXX. In particular, these volume forms are given as solutions of a nonlinear elliptic partial differential equation. As an application, we provide several examples ofdd-semistable SNC complex surfaces with trivial canonical bundle including double curves, which are smoothable to complex tori, primary Kodaira surfaces, andK3K3surfaces. We also provide several examples of such complex surfaces including triple points, which are smoothable toK3K3surfaces.

Numerical solution of second-order nonlinear partial differential equations originating from physical phenomena using Hermite based block methods

A Hermite based block method (HBBM) is proposed for the numerical solution of second-order non-linear elliptic partial differential equations (PDEs). The development of the method was accomplished through the methodology of interpolation and collocation procedures. The method’s analysis reveals that it satisfied the requirements for a numerical technique to be convergent. The implementation of the method is extensively discussed. Five numerical examples originating from physical phenomena are presented, and the applicability and accuracy of the HBBM are established by comparing them with the existing methods; the haar wavelet collocation method, the modified cubic B-spline collocation method, and the modified decomposition method. The proposed methods of HBBM are more accurate, stable, and convergent

A Complete Procedure for a Constraint-Type Fictitious Time Integration Method to Solve Nonlinear Multi-Dimensional Elliptic Partial Differential Equations

In this paper, an efficient and straightforward numerical procedure is constructed to solve multi-dimensional linear and nonlinear elliptic partial differential equations (PDEs). Although the numerical procedure for the constraint-type fictitious time integration method overcomes the numerical stability problem, the parameter’s definition, numerical accuracy and computational efficiency have not been resolved, and the lack of initial guess values results in reduced computational efficiency. Therefore, the normalized two-point boundary value solution of the Lie-group shooting method is proposed and considered in the numerical procedure to avoid the problem of the initial guess value. Then, a space-time variable, including the minimal fictitious time step and convergence rate factor, is introduced to study the relationship between the initial guess value and convergence rate factor. Some benchmark numerical examples are tested. As the results show, this numerical procedure using the normalized boundary value solution can significantly converge within one step, and the numerical accuracy is better than that demonstrated in the previous literature.

Boundary Lower Estimates for a Class of Fully Nonlinear Elliptic Partial Differential Equations

In this paper we are concerned with the Dirichlet problem for a class of fully nonlinear elliptic partial differential equations. Under structural conditions, we establish boundary estimates for the lower bound of convex solutions to the problem. There are two major ingredients in our proof: one is the introduction of an (a, η, μ) type condition, which serves as an important local condition of convex domains and better adapts to the study of boundary lower estimates; the other is the construction of super-solutions, which involves using the convex envelope of auxiliary functions over convex subdomains. As an application, we show the optimal boundary lower estimates of convex solutions to hyperbolic affine sphere equations.

A high-resolution fuzzy transform combined compact scheme for 2D nonlinear elliptic partial differential equations

This paper proposes a new high-resolution fuzzy transform algorithm for solving two-dimensional nonlinear elliptic partial differential equations (PDEs). The underlying new computational method implements the method of so-called approximating fuzzy components, which evaluate the solution values with fourth-order accuracy at internal mesh points. Triangular basic functions and fuzzy components are locally determined by linear combinations of solution values at nine points. Such a scheme connects the proposed method of approximating fuzzy components with the exact values of the solution using a linear system of equations. Compact approximations of high-resolution fuzzy components using nine points give a block tridiagonal Jacobi matrix. Apart from the numerical solution, it is easy to construct closed-form approximate solutions using a 2D spline interpolation polynomial from the available data with fuzzy components. The upper bounds of the approximation errors are estimated, as well as the convergence of the approximating solutions. Simulations with linear and nonlinear elliptical PDEs arising from quantum mechanics and convection-dominated diffusion phenomena are presented to confirm the usefulness of the new scheme and fourth-order convergence. To summarize:•The paper presents a high-resolution numerical method for the two-dimensions elliptic PDEs with nonlinear terms.•The combined effect of fuzzy transform and compact discretizations yields almost fourth-order accuracies to Schro¨dinger equation, convection-diffusion equation, and Burgers equation.•The high-order numerical scheme is computationally efficient and employs minimal data storage.

Local boundedness of weak solutions to elliptic equations with $ p, q- $growth

&lt;abstract&gt;&lt;p&gt;This article is dedicated to Giuseppe Mingione for his $ 50^{th} $ birthday, a leading expert in the regularity theory and in particular in the subject of this manuscript. In this paper we give conditions for the &lt;italic&gt;local boundedness&lt;/italic&gt; of weak solutions to a class of nonlinear elliptic partial differential equations in divergence form of the type considered below in (1.1), under $ p, q- $growth assumptions. The novelties with respect to the mathematical literature on this topic are the general growth conditions and the explicit dependence of the differential equation on $ u $, other than on its gradient $ Du $ and on the $ x $ variable.&lt;/p&gt;&lt;/abstract&gt;

A one-shot overlapping Schwarz method for component-based model reduction: application to nonlinear elasticity

We propose a component-based (CB) parametric model order reduction (pMOR) formulation for parameterized nonlinear elliptic partial differential equations (PDEs) based on overlapping subdomains. Our approach reads as a constrained optimization statement that penalizes the jump at the components’ interfaces subject to the approximate satisfaction of the PDE in each local subdomain. Furthermore, the approach relies on the decomposition of the local states into a port component – associated with the solution on interior boundaries – and a bubble component that vanishes at ports: since the bubble components are uniquely determined by the solution value at the corresponding port, we can recast the constrained optimization statement into an unconstrained statement, which reads as a nonlinear least-squares problem and can be solved using the Gauss–Newton method. We present thorough numerical investigations for a two-dimensional neo-Hookean nonlinear mechanics problem to validate our method; we further discuss the well-posedness of the mathematical formulation and the a priori error analysis for linear coercive problems.

Partial Differential Equations

The workshop covered topics in nonlinear elliptic and parabolic Partial Differential Equations as well as topics in Geometric Measure Theory, touching topics such as geometric variational problems and minimal surfaces, geometric flows, free boundaries and the structure of nodal sets of eigenfunctions as well as real and complex Monge-Amp\`ere equations.

Local Lipschitz continuity for [formula omitted]PDEs with explicit [formula omitted]dependence

This article is dedicated to Emmanuele Di Benedetto, great mathematician, colleague, friend. In the spirit to treat a subject that in the last years attracted the interest of several mathematicians, and the attention of Emmanuele too, in this paper we give a first approach to the local Lipschitz continuity of weak solutions to a class of nonlinear elliptic partial differential equations in divergence form of the type ∑i=1n∂∂xiaix,u,Du=bx,u,Duunder p,q−growth assumptions. The novelties with respect to the mathematical literature on this topic are the general growth conditions and the explicit dependence of the differential equation on u, other than on its gradient Du and on the x variable.