Second-order Elliptic Equations Research Articles

Convergence analysis of deep Ritz method with over-parameterization.

The deep Ritz method (DRM) has recently been shown to be a simple and effective method for solving PDEs. However, the numerical analysis of DRM is still incomplete, especially why over-parameterized DRM works remains unknown. This paper presents the first convergence analysis of the over-parameterized DRM for second-order elliptic equations with Robin boundary conditions. We demonstrate that the convergence rate can be controlled by the weight norm, regardless of the number of parameters in the network. To this end, we establish novel approximation results in Sobolev spaces with norm constraints, which have independent significance.

EXISTENCE OF A RENORMALIZED SOLUTION OF A QUASI-LINEAR ELLIPTIC EQUATION WITHOUT THE SIGN CONDITION ON THE LOWEST TERM

The paper considers a second-order quasilinear elliptic equation with an integrable right-hand side. Restrictions on the structure of the equation are formulated in terms of the generalized 𝑁-function. Unlike the author’s previous works, there is no sign condition for the low-order term of the equation. In non-reflexive Musielak–Orlicz–Sobolev spaces in an arbitrary unbounded strictly Lipschitz domain, the existence of a renormalized solution to the Dirichlet problem of this equation is proven.

Asymptotic homogenization of a mechanical equilibrium problem of a functionally-graded Euler-Bernoulli beam with non-periodic microstructure

To the best of our knowledge, the few classical applications of Keller's two-space method of non-periodic asymptotic homogenization are related to the effective behavior of heterogeneous media in the context of poroelasticity considering fluid flow and saturation. We believe that this is due to the alternative common approach of approximating random or non-periodic microstructures via the periodic replication of a representative volume element, as periodic structures are, generally speaking, much more tractable mathematically and computationally. However, more than 40 years later, a number of preliminary results of applications of the two-space method has arisen on various areas, namely, effective behavior of composite or functionally-graded bars, approximate solution of the electroencephalogram forward problem for neural imaging activity, and modeling of atmospheric pollutant dispersion. These recent applications deal with second-order elliptic or parabolic equations. In this contribution, we present the application of the two-space method to a mechanical equilibrium problem of a functionally-graded Euler-Bernoulli beam with non-periodic microstructure, which relies on a fourth-order elliptic equation. To the best of our knowledge, homogenization of fourth-order equations has been considered only in the periodic case.

On the Boyarsky–Meyers Estimate for the Solution of the Dirichlet Problem for a Second-Order Linear Elliptic Equation with Drift

On the Boyarsky–Meyers Estimate for the Solution of the Dirichlet Problem for a Second-Order Linear Elliptic Equation with Drift

Structural stability of non-isentropic Euler-Poisson system for gaseous stars

This paper concerns the non-isentropic steady compressible Euler-Poisson system in annuluses, which models the motion of gaseous stars with the gravitational interactions between gas particles and pressure forces. In the paper, the Euler-Poisson system is reformulated and decomposed into transport equations and coupled second-order nonlinear elliptic equations in polar coordinates. Not only the existence and the uniqueness, but also the structural stability of subsonic solutions are established.

Global Hessian estimate for second-order elliptic equation in Hardy spaces

Global Hessian estimate for second-order elliptic equation in Hardy spaces

Existence and non-existence results for a singular second-order elliptic equation

Using the variational methods, this paper is concerned with the existence and non-existence results to a class of elliptic equation with singular term on compact Riemannian manifold. In the context of complete, non-compact and non parabolic manifold, we obtain the non-existence result of solution for this type of elliptic problems.

Solution of the Elliptic Interface Problem by a Hybrid Mixed Finite Element Method

This paper addresses the elliptic interface problem involving jump conditions across the interface. We propose a hybrid mixed finite element method on the triangulation where the interfaces are aligned with the mesh. The second-order elliptic equation is initially decomposed into two equations by introducing a gradient term. Subsequently, weak formulations are applied to these equations. Scheme continuity is enforced using the Lagrange multiplier technique. Finally, we derive an explicit formula for the entries of the matrix equation representing Lagrange multiplier unknowns resulting from hybridization. The method yields approximations of all variables, including the solution and gradient, with optimal order. Furthermore, the matrix representing the final linear algebra systems is not only symmetric but also positive definite. Numerical examples convincingly demonstrate the effectiveness of the hybrid mixed finite element method in addressing elliptic interface problems.

Fourth-Order Accurate Compact Scheme for First-Order Maxwell’s Equations

We construct a compact fourth-order scheme, in space and time, for the time-dependent Maxwell’s equations given as a first-order system on a staggered (Yee) grid. At each time step, we update the fields by solving positive definite second-order elliptic equations. We develop compatible boundary conditions for these elliptic equations while maintaining a compact stencil. The proposed scheme is compared computationally with a non-compact scheme and with a convolutional dispersion relation preserving (DRP) scheme.

Singular solutions for complex second order elliptic equations and their application to time-harmonic diffuse optical tomography

We construct singular solutions of a complex elliptic equation of second order, having an isolated singularity of any order. In particular, we extend results obtained for the real partial differential equation in divergence form by Alessandrini in 1990. Our solutions can be applied to the determination of the optical properties of an anisotropic medium in time-harmonic Diffuse Optical Tomography (DOT).

Analytic and Gevrey class regularity for parametric semilinear reaction-diffusion problems and applications in uncertainty quantification

We investigate a class of parametric elliptic semilinear partial differential equations of second order with homogeneous essential boundary conditions, where the coefficients and the right-hand side (and hence the solution) may depend on a parameter. This model can be seen as a reaction-diffusion problem with a polynomial nonlinearity in the reaction term. The efficiency of various numerical approximations across the entire parameter space is closely related to the regularity of the solution with respect to the parameter. We show that if the coefficients and the right-hand side are analytic or Gevrey class regular with respect to the parameter, the same type of parametric regularity is valid for the solution. The key ingredient of the proof is the combination of the alternative-to-factorial technique from our previous work [1] with a novel argument for the treatment of the power-type nonlinearity in the reaction term. As an application of this abstract result, we obtain rigorous convergence estimates for numerical integration of semilinear reaction-diffusion problems with random coefficients using Gaussian and Quasi-Monte Carlo quadrature. Our theoretical findings are confirmed in numerical experiments.

Some auxiliary estimates for solutions to non-uniformly degenerate second-order elliptic equations

We consider a class of second order elliptic equations in divergence form with non-uniform exponential degeneracy. The method used is based on the fact that the degeneracy rates of the eigenvalues of the matrix ||aij(x)|| (function λi(x)) are not the functions of unusual norm |x|, but of some anisotropic distance |x| a−. We assume that the Dirichlet problem for such equations is solvable in the classical sense for every continuous boundary function in any normal domain Ω. Estimates for the weak solutions of Dirichlet problem near the boundary point are obtained, and Green’s functions for second order non-uniformly degenerate elliptic equations are constructed.

On the Boyarsky–Meyers Estimate for the Gradient of the Solution to the Dirichlet Problem for a Second-Order Linear Elliptic Equation with Drift: The Case of Critical Sobolev Exponent

On the Boyarsky–Meyers Estimate for the Gradient of the Solution to the Dirichlet Problem for a Second-Order Linear Elliptic Equation with Drift: The Case of Critical Sobolev Exponent

On the Boyarsky-Meyers estimate for the solution of the Dirichlet problem for a second-order linear elliptic equation with drift

We establish the increased integrability of the gradient of the solution to the Dirichlet problem for the Laplace operator with lower terms and prove the unique solvability of this problem.

Robust domain decomposition methods for high-contrast multiscale problems on irregular domains with virtual element discretizations

Our research focuses on the development of domain decomposition preconditioners tailored for second-order elliptic partial differential equations. Our approach addresses two major challenges simultaneously: i) effectively handling coefficients with high-contrast and multiscale properties, and ii) accommodating irregular domains in the original problem, the coarse mesh, and the subdomain partition. The robustness of our preconditioners is crucial for real-world applications, such as the efficient and accurate modeling of subsurface flow in porous media and other important domains.The core of our method lies in the construction of a suitable partition of unity functions and coarse spaces utilizing local spectral information. Leveraging these components, we implement a two-level additive Schwarz preconditioner. We demonstrate that the condition number of the preconditioned systems is bounded with a bound that is independent of the contrast. Our claims are further substantiated through selected numerical experiments, which confirm the robustness of our preconditioners.

Numerical Solutions of Second-Order Elliptic Equations with C-Bézier Basis

Numerical Solutions of Second-Order Elliptic Equations with C-Bézier Basis

The Regularity of Solutions to Mixed Boundary Value Problems of Second-Order Elliptic Equations with Small Angles

The Regularity of Solutions to Mixed Boundary Value Problems of Second-Order Elliptic Equations with Small Angles

ON ONE ELLIPTIC EQUATION WITH UNBOUNDED COEFFICIENTS

Abstract. The paper studies a second-order elliptic equation given on an infinite strip of width 2 and having variable intermediate and junior coefficients. We assume that the intermediate coefficients may not obey the potential. The problem is posed of finding a strong generalized solution of the equation that satisfies periodic conditions at the edges of the strip. The coefficients of the equation are smooth and not bounded functions. In terms of the coefficients themselves, the sufficient conditions for the existence and uniqueness of a solution to the problem posed are indicated, and a maximal regularity estimate for the solution is given. Their main difference from known results is the requirement of a different order of growth of intermediate coefficients for partial derivatives of an unknown function with respect to different variables. We also cover the case where the intermediate coefficients can grow at infinity faster than a linear function.

High-order estimates for fully nonlinear equations under weak concavity assumptions

This paper studies a priori and regularity estimates of Evans-Krylov type in Hölder spaces for fully nonlinear uniformly elliptic and parabolic equations of second order when the operator fails to be concave or convex in the space of symmetric matrices. In particular, it is assumed that either the level sets are convex or the operator is concave, convex or close to a linear function near infinity. As a byproduct, these results imply polynomial Liouville theorems for entire solutions of elliptic equations and for ancient solutions to parabolic problems.

On Dirichlet problem and uniform approximation by solutions of second-order elliptic systems in [formula omitted

We consider the Dirichlet problem for solutions of second-order elliptic systems in the plane that correspond to homogeneous elliptic equations with constant complex coefficients. We prove that any Jordan domain G⊂C with C1,α-smooth boundary, 0<α<1, is not regular with respect to the Dirichlet problem for any non-strongly elliptic equation under consideration, which means that there always exists a continuous complex valued function on ∂G that can not be continuously extended to G to a function satisfying the corresponding equation therein. Since there exists a Jordan domain with Lipschitz boundary, which is regular with respect to the Dirichlet problem for bianalytic functions, the result obtained is near to be sharp. We also consider the problem on uniform approximation of functions by polynomial solutions of homogeneous second-order elliptic equations with constant complex coefficients and give a new proof of the approximation criterion in this problem on Carathéodory compact sets.