Second-order Elliptic Equations Research Articles

Stability and instability results for sign-changing solutions to second-order critical elliptic equations

On a smooth, closed Riemannian manifold (M,g) of dimension n≥3, we consider the stationary Schrödinger equation Δgu+h0u=|u|2⁎−2u, where Δg:=−divg∇, h0∈C1(M) and 2⁎:=2nn−2. We prove that, up to perturbations of the potential function h0 in C1(M), the sets of sign-changing solutions that are bounded in H1(M) are precompact in the C2 topology. We obtain this result under the assumptions that (M,g) is locally conformally flat, n≥7 and h0≠n−24(n−1)Scalg at all points in M, where Scalg is the scalar curvature of the manifold. We then provide counterexamples in every dimension n≥3 showing the optimality of these assumptions.

An adaptive stochastic Galerkin method based on multilevel expansions of random fields: Convergence and optimality

The subject of this work is a new stochastic Galerkin method for second-order elliptic partial differential equations with random diffusion coefficients. It combines operator compression in the stochastic variables with tree-based spline wavelet approximation in the spatial variables. Relying on a multilevel expansion of the given random diffusion coefficient, the method is shown to achieve optimal computational complexity up to a logarithmic factor. In contrast to existing results, this holds in particular when the achievable convergence rate is limited by the regularity of the random field, rather than by the spatial approximation order. The convergence and complexity estimates are illustrated by numerical experiments.

Hermite Fitted Block Integrator for Solving Second-Order Anisotropic Elliptic Type PDEs

A Hermite fitted block integrator (HFBI) for numerically solving second-order anisotropic elliptic partial differential equations (PDEs) was developed, analyzed, and implemented in this study. The method was derived through collocation and interpolation techniques using the Hermite polynomial as the basis function. The Hermite polynomial was interpolated at the first two successive points, while the collocation occurred at all the suitably chosen points. The major scheme and its complementary scheme were united together to form the HFBI. The analysis of the HFBI showed that it had a convergence order of eight with small error constants, was zero-stable, absolutely-stable, and satisfied the condition for convergence. In order to confirm the usefulness, accuracy, and efficiency of the HFBI, the method of lines approach was applied to discretize the second-order anisotropic elliptic partial differential equation PDE into a system of second-order ODEs and consequently used the derived HFBI to obtain the approximate solutions for the PDEs. The computed solution generated by using the HFBI was compared to the exact solutions of the problems and other existing methods in the literature. The proposed method compared favorably with other existing methods, which were validated through test problems whose solutions are presented in tabular form, and the comparisons are illustrated in the curves.

Optimal [formula omitted](div) flux approximations from the Primal Hybrid Finite Element Method on quadrilateral meshes

In this work, we propose and analyze an element-wise H(div,Ω)-conforming computational strategy to post-process the flux vector from the approximated solution of a second-order elliptic equation, obtained by the Primal Hybrid Finite Element Method on regular meshes of convex quadrilaterals generated by bilinear isomorphisms. The recovery strategy makes use of the Arnold–Boffi–Falk (ABFt, t≥0) spaces, leading to locally conservative approximated fluxes with continuous normal components. The proposed method is proven to provide optimal order approximations in H(div,Ω), in the sense that both the flux and its divergence achieve optimal order ht+1 in the L2(Ω)-norm, on quadrilateral meshes. We also show that, within the recovery strategy, it is possible to achieve higher-order approximations for the divergence of the flux by increasing the index t of the local spaces, with almost no extra overall computational cost. Numerical results on affine and bilinear quadrilateral meshes are presented, confirming the theoretical predictions.

Analysis of a weak Galerkin method for second-order elliptic equations with minimal regularity on polytopal meshes

We propose an error analysis of a weak Galerkin method for second-order elliptic equations with minimal regularity requirements, i.e., the exact solution u belongs to H1+s(Ω) with s>0 possibly close to zero. In this case, the conormal derivative of u is not well-defined on the mesh faces, which results in a legitimacy problem of the error equation originally derived under smooth assumption. We employ the weak meaning of the conormal derivative introduced in [Ern, A., & Guermond, J. L. (2021). Quasi-optimal Nonconforming Approximation of Elliptic PDEs with Contrasted Coefficients andH1+s,s>0, Regularity. Foundations of Computational Mathematics, 1–36.] to establish new error equation and optimal a priori error estimates for low regular solutions on polytopal meshes. We also investigate the corresponding elliptic eigenvalue problems. The numerical experiments verify the theoretical results and agree with the benchmark eigenvalue problem.

On some properties of elliptic partial differential equation solutions

This paper is devoted to the study of properties of second-order elliptic equation solutions. The main content of the paper coincides with the report made by the author at the international conference dedicated to the 75th anniversary of I. V. Volovich. The solution behavior near the boundary and the Dirichlet problem formulation, which is closely related to this issue, are studied. At the end of the paper, we will briefly discuss the results obtained in elegant and extremely important works by E. De Giorgi and J. Nash regarding Hölder continuity of the equation solutions within the considered domain. We present results that combine and complement the belonging of the solution to the Hölder and Sobolev spaces. Note that all the concepts and statements under consideration are united by a common approach and are formulated in close terms.

Maximal L1 regularity for solutions to inhomogeneous incompressible Navier-Stokes equations

This paper is devoted to the maximal L1 regularity and asymptotic behavior for solutions to the inhomogeneous incompressible Navier-Stokes equations under a scaling-invariant smallness assumption on the initial velocity. We obtain a new global L1-in-time estimate for the Lipschitz seminorm of the velocity field without any smallness assumption on the initial density fluctuation. In the derivation of this estimate, we study the maximal L1 regularity for a linear Stokes system with variable coefficients. The analysis tools are a use of the semigroup generated by a generalized Stokes operator to characterize some Besov norms and a new gradient estimate for a class of second-order elliptic equations of divergence form. Our method can be used to study some other issues arising from incompressible or compressible viscous fluids.

Stability of stationary subsonic compressible Euler flows with mass-additions in two-dimensional straight ducts

We show existence, uniqueness and stability for a family of stationary subsonic compressible Euler flows with mass-additions in two-dimensional rectilinear ducts, subjected to suitable time-independent multi-dimensional boundary conditions at the entrances and exits. The stationary subsonic Euler equations consist a quasi-linear system of elliptic-hyperbolic composite-mixed type, while addition-of-mass destructs the usual methods based upon conservation of mass and Lagrangian coordinates to separate the elliptical and hyperbolic modes of the system. We establish a new decomposition and nonlinear iteration scheme to overcome this major difficulty. It reveals that mass-additions introduce very strong interactions in the elliptic and hyperbolic modes, and lead to a class of second-order elliptic equations with multiple integral nonlocal terms. The linearized problem is solved by studying algebraic and analytical properties of infinite weakly coupled boundary-value problems of ordinary differential equations, each with multiple nonlocal terms, after applications of Fourier analysis methods.

Global gradient estimates for Dirichlet problems of elliptic operators with a BMO antisymmetric part

Abstract Let n ≥ 2 n\ge 2 and Ω ⊂ R n \Omega \subset {{\mathbb{R}}}^{n} be a bounded nontangentially accessible domain. In this article, the authors investigate (weighted) global gradient estimates for Dirichlet boundary value problems of second-order elliptic equations of divergence form with an elliptic symmetric part and a BMO antisymmetric part in Ω \Omega . More precisely, for any given p ∈ ( 2 , ∞ ) p\in \left(2,\infty ) , the authors prove that a weak reverse Hölder inequality with exponent p p implies the global W 1 , p {W}^{1,p} estimate and the global weighted W 1 , q {W}^{1,q} estimate, with q ∈ [ 2 , p ] q\in \left[2,p] and some Muckenhoupt weights, of solutions to Dirichlet boundary value problems. As applications, the authors establish some global gradient estimates for solutions to Dirichlet boundary value problems of second-order elliptic equations of divergence form with small BMO {\rm{BMO}} symmetric part and small BMO {\rm{BMO}} antisymmetric part, respectively, on bounded Lipschitz domains, quasi-convex domains, Reifenberg flat domains, C 1 {C}^{1} domains, or (semi-)convex domains, in weighted Lebesgue spaces. Furthermore, as further applications, the authors obtain the global gradient estimate, respectively, in (weighted) Lorentz spaces, (Lorentz–)Morrey spaces, (Musielak–)Orlicz spaces, and variable Lebesgue spaces. Even on global gradient estimates in Lebesgue spaces, the results obtained in this article improve the known results via weakening the assumption on the coefficient matrix.

Fully Nonlinear Elliptic Equations on Hermitian Manifolds for Symmetric Functions of Partial Laplacians

We consider a class of fully nonlinear second-order elliptic equations on Hermitian manifolds closely related to the general notion of $$\mathbb {G}$$ -plurisubharmonicity of Harvey–Lawson and an equation treated by Székelyhidi–Tosatti–Weinkove in the proof of Gauduchon conjecture. Under fairly general assumptions, we derive interior estimates and establish the existence of smooth solutions for the Dirichlet problem as well as for equations on closed manifolds.

Asymptotic mean-value formulas for solutions of general second-order elliptic equations

Abstract We obtain asymptotic mean-value formulas for solutions of second-order elliptic equations. Our approach is very flexible and allows us to consider several families of operators obtained as an infimum, a supremum, or a combination of both infimum and supremum, of linear operators. The families of equations that we consider include well-known operators such as Pucci, Issacs, and k-Hessian operators.

Existence of solutions of nonlinear elliptic equations with measure data in Musielak-Orlicz spaces

Abstract A second-order quasilinear elliptic equation with a measure of special form on the right-hand side is considered. Restrictions on the structure of the equation are imposed in terms of a generalized -function such that the conjugate function obeys the -condition and the corresponding Musielak-Orlicz space is not necessarily reflexive. In an arbitrary domain satisfying the segment property, the existence of an entropy solution of the Dirichlet problem is proved. It is established that this solution is renormalized. Bibliography: 29 titles.

Adaptive two-layer ReLU neural network: II. Ritz approximation to elliptic PDEs

In the companion paper [1], we introduce adaptive network enhancement (ANE) method for the best least-squares approximation to a target function by using two-layer ReLU neural networks (NNs). In this paper, we apply the ANE method for solving self-adjoint second-order elliptic partial differential equations (PDEs). The underlying PDE is discretized by the Ritz method using a two-layer spline neural network based on either the primal or dual formulations that minimize the respective energy or complimentary functionals. Essential boundary conditions are imposed weakly through the functionals with proper norms. It is proved that the Ritz approximation is the best approximation in the energy norm; moreover, effect of numerical integration for the Ritz approximation is analyzed as well. Two estimators for adaptive neuron enhancement method are introduced, one is the so-called recovery estimator and the other is the least-squares estimator. Finally, numerical results for diffusion problems with either corner or intersecting interface singularities are presented.

An adjoint-based adaptive error approximation of functionals by the hybridizable discontinuous Galerkin method for second-order elliptic equations

This paper presents a novel, fully computable error approximation and mesh adaptation approach for functionals defined by second-order elliptic equations. The functionals are approximated by the hybridizable discontinuous Galerkin (HDG) method and the error approximation is obtained by the adjoint-based method and a local post-processing technique of the HDG method. Unlike most adjoint-based error estimations in the literature, the novelty of our method is that the error approximation is obtained without requiring an auxiliary finer mesh or higher order approximation spaces for solving the adjoint problem. This reduces the computational cost and eases the implementation of the problem. What's more, the local post-processing technique can be carried out in parallel, which speeds up the method even more. We illustrate the method with a second-order elliptic problem and we present examples of three types of functionals: volume integrals; boundary integrals and eigenvalue problems. Numerical tests with adaptive mesh refinements for non-smooth solutions are presented to show that our method is efficient and robust.

A class of finite element methods with averaging techniques for solving the three-dimensional drift-diffusion model in semiconductor device simulations

Obtaining a satisfactory numerical solution of the classical three-dimensional drift-diffusion (DD) model, widely used in semiconductor device simulations, is still challenging nowadays, especially when the convection dominates the diffusion. In this work, we propose a series of finite element schemes with different types of averaging techniques to discretize the three-dimensional continuity equations. Our methods are based on the classical finite element framework, quite different from those mixed finite element/volume methods that also employ inverse averaging techniques. At first, the Slotboom variables are employed to transform the continuity equations into self-adjoint second-order elliptic equations with exponentially behaved coefficients. Then four averaging techniques, denoted with A1-A4, are introduced to approximate the exponential coefficient with its average on every tetrahedral element of the grid. The first scheme calculates the harmonic average of the exponential coefficient on a whole tetrahedron, and the other three schemes calculate the average of the exponential coefficient on each edge of a tetrahedral element. Our methods can avoid the spurious non-physical numerical oscillations and guarantee the conservation of the computed terminal currents with a terminal current evaluation approach. In fact, these methods not only maintain numerical stability but also overcome the disadvantages of some stabilization methods that cannot guarantee the conservation of the terminal currents, such as the streamline-upwind Petrov-Galerkin (SUPG) method. Moreover, the derivation of these discretization methods does not need the dual Voronoi grid as that of the finite volume Scharfetter-Gummel (FVSG) method or other mixed finite element/volume methods with inverse averaging techniques, which greatly reduces the complexity of parallel implementations of our methods. Simulations on two realistic three-dimensional semiconductor devices are carried out to test the accuracy and stability of our methods. According to numerical results, we conclude that scheme A4 can produce more accurate numerical solutions than the other three schemes, especially when the bias applied on the electrode is high. Numerical results also show that the scheme A4 is more robust than the Zlámal finite element method [1] in high-bias cases, and it also performs better than the FVSG method and a tetrahedral mixed finite element method [2] on poor-quality grids. Scheme A4 is also employed to study rich physical properties of the n-channel MOSFET.

Oblique derivative problem in a plane sector for strong quasi-linear elliptic second-order equation with perturbed [formula omitted]-Laplacian

We study the behavior near the boundary corner point of bounded weak solutions to the oblique derivative problem for singular p(x)-Laplacian equation with strong nonlinearities singular absorption term in a bounded plane sector.

Stochastic Algorithms for Solving the Dirichlet Boundary Value Problem for Certain Second-Order Elliptic Equations with Discontinuous Coefficients

Stochastic Algorithms for Solving the Dirichlet Boundary Value Problem for Certain Second-Order Elliptic Equations with Discontinuous Coefficients

Unified discontinuous Galerkin scheme for a large class of elliptic equations

We present a discontinuous Galerkin internal-penalty scheme that is applicable to a large class of linear and nonlinear elliptic partial differential equations. The unified scheme can accommodate all second-order elliptic equations that can be formulated in first-order flux form, encompassing problems in linear elasticity, general relativity, and hydrodynamics, including problems formulated on a curved manifold. It allows for a wide range of linear and nonlinear boundary conditions, and accommodates curved and nonconforming meshes. Our generalized internal-penalty numerical flux and our Schur-complement strategy of eliminating auxiliary degrees of freedom make the scheme compact without requiring equation-specific modifications. We demonstrate the accuracy of the scheme for a suite of numerical test problems. The scheme is implemented in the open-source SpECTRE numerical relativity code.

Gradient estimate for solutions of second-order elliptic equations

We obtain a local estimate for the gradient of solutions to a second-order elliptic equation in divergence form with bounded measurable coefficients that are square-Dini continuous at the single point $x=0$. In particular, we treat the case of solutions that are not Lipschitz continuous at $x=0$. We show that our estimate is sharp.

A Continuous Five-step Implicit Block Unification Method for Numerical Solution of Second-Order Elliptic Partial Differential Equations (PDEs)

A Continuous Five-step Implicit Block Unification Method for Numerical Solution of Second-Order Elliptic Partial Differential Equations (PDEs)