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Based on partial Fourier series analysis, we adapt on a model case a new approach to classical results obtained in the literature describing the singularities of a family a solutions of a second‐order elliptic problems on open non‐convex planar sectors. The method allows the exhibition of singular and regular frequencies, explicit decomposition, and description of coefficients of singularities of the solution. As a main result, explicit and sharp estimates with respect to the opening angle parameter are obtained via this method. They are not uniform near π$$ \pi $$ where corners have opening angle generating a jump of singularity in Sobolev exponent, contrarily to the results obtained for harmonic and/or biharmonic problems on a family of convex planar sectors. [ABSTRACT FROM AUTHOR]

We consider an elliptic problem with nonlinear boundary condition involving nonlinearity with superlinear and subcritical growth at infinity and a bifurcation parameter as a factor. We use re-scaling method, degree theory and continuation theorem to prove that there exists a connected branch of positive solutions bifurcating from infinity when the parameter goes to zero. Moreover, if the nonlinearity satisfies additional conditions near zero, we establish a global bifurcation result, and discuss the number of positive solution(s) with respect to the parameter using bifurcation theory and degree theory. [ABSTRACT FROM AUTHOR]

The present paper is devoted to study the asymptotic behavior of a sequence of linear elliptic equations with a varying drift term, whose coefficients are just bounded in L N (Ω) , with N the dimension of the space. It is known that there exists a unique solution for each of these problems in the Sobolev space H 0 1 (Ω). However, because the operators are not coercive, there is no uniform estimate of the solutions in this space. We use some estimates in (J. Differential Equations258 (2015) 2290–2314), and a regularization obtained by adding a small nonlinear first order term, to pass to the limit in these problems. [ABSTRACT FROM AUTHOR]

This study investigates existence result of a nontrivial weak solutions for a system of nonlinear elliptic equations involving Hardy Potentials. The problem arises in the mathematical modeling of complex physical phenomena. [ABSTRACT FROM AUTHOR]

In this article, we study some anisotropic singular perturbations for a class of linear elliptic problems. A uniform estimates for conforming $ Q_1 $ Q 1 finite element method are derived, and some other results of convergence and regularity for the continuous problem are proved. The arguments are based on the tensorization trick and standard elliptic regularity theory tools. [ABSTRACT FROM AUTHOR]

The novelty of this note is to establish existence result for the following anisotropic elliptic equation ... where the datum f ∈ L¹ (Ω) and ... and H (x,ϑ) ∈ L¹ (Ω). Furthermore only the large monotonicity conditions will be assumed on B(x; s; ξ). To overcome this difficulty we will use the approach of Minty's lemma in the anisotropic weighted Sobolev spaces. [ABSTRACT FROM AUTHOR]

In this paper, we study a linear elliptic system involving Navier-type boundary conditions or tangential components in a three-dimensional bounded domain, possibly multiply connected and with boundary possibly nonconnected. We establish an appropriate Inf-Sup condition and then we prove the existence and uniqueness of the generalized solutions in Lp-theory. We also investigate the case of strong solutions. [ABSTRACT FROM AUTHOR]

In this paper, we are concerned with the following critical fractional Choquard equation$ (-\Delta)^{s}u + V(x)u = (\mathcal{K}_{\mu}*|u|^{2^{*}_{\mu,s}})|u|^{2^{*}_{\mu,s}-2}u, \ \ u \in D^{s,2}(\mathbb{R}^{N}), $where $ s\in (0,1) $, $ N\geq 3 $, $ \max\{N-4s, 0\}<\mu

In this article, the existence and nonexistence of solutions for the quasilinear elliptic equations involving multiple critical terms under Dirichlet boundary conditions on bounded smooth domains Ω⊂RN(N≥3)\Omega \subset {R}^{N}(N\ge 3) are proved by using the variational method and Pohozaev identity, respectively.

In this paper we study the existence of solutions of a one-dimensional eigenvalue problem − (| ϕ x | p − 2 ϕ x) x = λ (| ϕ | q − 2 ϕ − f (ϕ)) such that ϕ (0) = ϕ (1) = 0 , where p , q > 1 , λ is a positive real parameter and f is a continuous (not necessarily odd) function. Our goal is to give a complete description of solutions of this problem. We completely characterize the set of solutions of this problem, which may be uncountable. For 1 < p ≠ 2 , the existing results treat only the case when f is either odd and a power (see [11]) or when p = q ([8]). Our method of proof relies on a careful analysis of the phase diagram associated with this equation, refining the regularity results of M. Ôtani in 1984 (see [10]) and characterizing the exact points where we may have C 2 regularity of solutions including some points χ ∈ (0 , 1) for which ϕ x (χ) = 0. [ABSTRACT FROM AUTHOR]

In this paper we show existence of positive solutions for a class of problems involving the Choquard term in exterior domain and the nonlinearity with critical growth. We prove the main results using variational method combined with Brouwer theory of degree and Deformation lemma. [ABSTRACT FROM AUTHOR]

Computing the distance function to some surface or line is a problem that occurs very frequently. There are several ways of computing a relevant approximation of this function, using for example technique originating from the approximation of Hamilton Jacobi problems, or the fast sweeping method. Here we make a link with some elliptic problem and propose a very fast way to approximate the distance function.

In this article, we examine a higher order convergent approximation for a class of singularly perturbed two-dimensional (2-D) convection-diffusion-reaction elliptic problems with discontinuous convection and source terms. The coefficient of the highest-order derivative of this problem, is considered to be arbitrarily small. The solutions of these problems exhibit interior layers due to the discontinuity in the convection and source terms. In this article, a numerical approach is proposed by using a hybrid-difference technique (a suitable combination of the midpoint upwind technique in the smooth regions and the central difference technique in the layer regions) on a propitious layer-adaptive piecewise-uniform Shishkin mesh which leads to an almost second-order estimate. We present numerical experiments with several combinations of possible discontinuities to verify that our proposed results are sharp. [ABSTRACT FROM AUTHOR]

In this article, we present a physics-informed neural network combined with fictitious domain method (FDM-PINN) to study linear elliptic and parabolic problems with Robin boundary condition. Our goal here to develop a deep learning framework where one solves a variant of the original problem on the full Ω, followed by a well-chosen correction on small domain ω (ω ¯ ⊂ Ω) , which is geometrically simple shaped domain. We study the applicability and accuracy of FDM-PINN for the elliptic and parabolic problems with fixed ω or moving ω. This method is of the virtual control type and relies on a well-designed neural network. Numerical results obtained by FDM-PINN for two-dimensional elliptic and parabolic problems are given, which are more accurate than the results obtained by least-squares/fictitious domain method in [R. Glowinski and Q. He, A least-squares/fictitious domain method for linear elliptic problems with robin boundary conditions, Commun. Comput. Phys. 9 (2011), pp. 587–606.]. [ABSTRACT FROM AUTHOR]

The positivity preservation is very important in most applications solving elliptic problems. Many schemes preserving positivity has been proposed but are at most second-order convergent. Besides, in general, high-order schemes do not preserve positivity. In the present paper, we propose an arbitrary-order positivity preserving method for elliptic problems in 2D. We show how to adapt our method to the case of a discontinuous and/or tensor-valued diffusion coefficient, while keeping the expected order of convergence. We assess the new scheme on several test problems. • First arbitrary-order positivity preserving finite volume scheme on deformed meshes. • Remains high-order accurate for discontinuous and/or tensor-valued diffusion coefficient. • Assessed on several numerical tests showing the order of convergence and the positivity. [ABSTRACT FROM AUTHOR]

A singularly perturbed elliptic problem, with non-smooth inflow or non-smooth outflow boundary conditions, is examined. A decomposition of the continuous solution is constructed, whose components identify the various types of layer functions that can exist in the solution. Parameter-explicit pointwise bounds on the first three partial derivatives of these components are established. An appropriate Shishkin mesh is identified for the problem and this is combined with upwinding to form a numerical method. Parameter-uniform error bounds are deduced. Numerical results are presented to illustrate the performance of the numerical method. [ABSTRACT FROM AUTHOR]

This paper deals with the existence of weak solutions for semilinear elliptic equation with nonlinearity on the boundary. We establish the existence of a maximal and a minimal weak solution between an ordered pair of sub- and supersolution for both monotone and nonmonotone nonlinearities. We use iteration argument when the nonlinearity is monotone. For the nonmonotone case, we utilize the surjectivity of a pseudomonotone and coercive operator, Zorn's lemma and a version of Kato's inequality.

Cross-points in domain decomposition, i.e., points where more than two subdomains meet, have received substantial attention over the past years, since domain decomposition methods often need special attention in their definition at cross-points, in particular if the transmission conditions of the domain decomposition method contain derivatives, like in the Dirichlet-Neumann method. We study here for the first time the convergence of the Dirichlet-Neumann method at the continuous level in the presence of cross-points. We show that its iterates can be uniquely decomposed into two parts, an even symmetric part that converges geometrically, like when there are no cross-points present, and an odd symmetric part, which generates a singularity at the cross-point and is not convergent. We illustrate our analysis with numerical experiments. [ABSTRACT FROM AUTHOR]

The purpose of this contribution it to theoretically analyze the adaptive refinement strategies for conforming hp-finite element approximations of elliptic problems proposed for exact algebraic solvers in (P. Daniel, A. Ern, I. Smears and M. Vohralk, Comput. Math. Appl. 76 (2018) 967–983.) and for inexact algebraic solvers in (P. Daniel, A. Ern, and M. Vohralk, Comput. Methods Appl. Mech. Eng. 359 (2020) 112607.). Both of these strategies are driven by guaranteed equilibrated flux energy error estimators. The employed hp-refinement criterion stems from solving two separate local residual problems posed only on the patches of elements around marked vertices selected by a bulk-chasing criterion. In the above references, we have derived a fully computable guaranteed bound on the ratio of the error on two successive steps of the hp-adaptive loop. Here, our focus is to prove that this ratio is uniformly smaller than one, and thus the convergence of the adaptive and adaptive inexact hp-refinement strategies. To be able to achieve this goal, we have to introduce some additional assumptions on the h- and p-refinements, namely an extension of the marked region, as well as a sufficient h- or p-refinement of each marked patch. We investigate two such strategies, where one ensures a polynomial-degree-robust guaranteed contraction. In the inexact case, a sufficiently precise stopping criterion for the algebraic solver is requested, but this criterion remains fully computable and also realistic in the sense that in our numerical experiments, it does not request the algebraic error to be excessively small in comparison with the total error. [ABSTRACT FROM AUTHOR]

We investigate the multiplicity of solutions for the one-dimensional elliptic Dirichlet boundary value problem with jumping nonlinearities. We obtain three theorems depending on the source terms when nonlinearities cross some eigenvalues. We obtain the first theorem and the second one by eigenvalues and the corresponding normalized eigenfunctions of the elliptic eigenvalue problem, and the contraction mapping principle on p-Lebesgue space. We obtain the third result by Leray–Schauder degree theory. [ABSTRACT FROM AUTHOR]

This paper deals with the existence of weak solutions for semilinear elliptic equation with nonlinearity on the boundary. We establish the existence of a maximal and a minimal weak solution between an ordered pair of sub- and supersolution for both monotone and nonmonotone nonlinearities. We use iteration argument when the nonlinearity is monotone. For the nonmonotone case, we utilize the surjectivity of a pseudomonotone and coercive operator, Zorn's lemma and a version of Kato's inequality. [ABSTRACT FROM AUTHOR]

This paper concerns the existence of a nontrivial solution for the following problem (P) - Δ u + V (x) u ∈ ∂ u F (x , u) a.e. in R N , u ∈ H 1 (R N). where F (x , t) = ∫ 0 t f (x , s) d s , f is a Z N -periodic measurable function and λ = 0 does not belong to the spectrum of −Δ + V. Here, ∂tF denotes the generalized gradient of F with respect to variable t. [ABSTRACT FROM AUTHOR]

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In this paper, we present a posteriori error estimator of edge residual-type Weak Galerkin mixed finite element method (WG-MFEM) solving second-order elliptic problems, where two different ways of a posteriori error estimator are presented, both of which hold on polygonal mesh. The two analyses are based on the standard FEM method and continuous local mass conservation property, respectively. In order to prove the reliability holding on polygonal meshes, the a posteriori error estimator is disposed by the post-processed numerical scalar solution. Compared with some other methods, the proposed a posteriori error estimator does not contain the post-processed solution and the efficiency can be obtained automatically. Numerical experiments are conducted to confirm the efficiency of this estimator. [ABSTRACT FROM AUTHOR]

In this note, we prove that the equal-area condition is necessary for the formation of internal transition layer by the solutions of a general singularly perturbed elliptic problem. Our result is independent of boundary conditions, it improves previous references in the literature and can be applied to Allen-Cahn, Sine-Gordon and Fisher-KPP equations, for example. [ABSTRACT FROM AUTHOR]

In this article, a hybrid scheme for a two‐parameter elliptic problem with regular exponential and boundary layers on Shishkin mesh is analyzed. The hybrid scheme comprises the central difference method in the layer region and the upwind method in the regular part. The use of the central difference in layer region results in a more accurate resolution of layers. The method is shown to have first‐order parameter uniform convergence. The numerical results corroborate the error estimates presented here. [ABSTRACT FROM AUTHOR]

In this paper, boundary fourth order semilinear elliptic problems involving nonnegative weight functions are investigated with indefinite asymptotically linear nonlinearities. Using the variational method, we prove the existence of nontrivial solutions without use of the Ambrosetti-Rabionovitz condition or any one of its replacements. When the non-linearities are superlinear at infinity, a suitable condition is added in order to use the same techniques to prove the existence of solutions. [ABSTRACT FROM AUTHOR]

This paper shows the unique solvability of elliptic problems associated with two-phase incompressible flows, which are governed by the two-phase Navier-Stokes equations with a sharp moving interface, in unbounded domains such as the whole space separated by a compact interface and the whole space separated by a non-compact interface. As a by-product, we obtain the Helmholtz-Weyl decomposition for two-phase incompressible flows. [ABSTRACT FROM AUTHOR]

We prove in weighted Orlicz-Sobolev spaces, the existence of entropy solution for a class of nonlinear elliptic equations of Leray-Lions type, with large monotonicity condition and right hand side f ∈ L1(Ω).

In this paper, a posteriori error estimates for the Weak Galerkin finite element methods (WG-FEMs) for second order elliptic problems are derived in terms of an H 1 − equivalent energy norm. Corresponding estimators based on the helmholtz decomposition yield globally upper and locally lower bounds for the approximation errors of the WG-FEMs. Especially, the error analysis of our methods is proved to be valid for polygonal meshes (e.g., hybrid, polytopal non-convex meshes and those with hanging nodes) under general assumptions. In addition, the work can make adaptive WG-FEMs solving partial differential equations such as stokes equations and biharmonic equations on polygonal meshes possible. Finally, we verify the theoretical findings by a few numerical examples. [ABSTRACT FROM AUTHOR]

Localized collocation methods based on radial basis functions (RBFs) for elliptic problems appear to be nonrobust in the presence of Neumann boundary conditions. In this paper, we overcome this issue by formulating the RBF-generated finite difference method in a discrete least squares setting instead. This allows us to prove high-order convergence under node refinement and to numerically verify that the least squares formulation is more accurate and robust than the collocation formulation. The implementation effort for the modified algorithm is comparable to that for the collocation method. [ABSTRACT FROM AUTHOR]

We consider the problem $$\displaylines{ \Delta_{p}u =|u|^{p-2}u \quad\text{in }\Omega, \cr |\nabla u|^{p-2}\frac{\partial u}{\partial \nu}=\lambda|u|^{p-2}u + g(\lambda,x,u) \quad\text{on }\partial\Omega, }$$ where $\Omega$ is a bounded domain of $\mathbb{R}^{N}$ with smooth boundary, $N\geq 2$, and $\Delta_p$ denotes the p-Laplacian operator. We give sufficient conditions for the existence of continua of solutions bifurcating from both zero and infinity at the principal eigenvalue of p-Laplacian with nonlinear boundary conditions. We also prove that those continua split on two, one containing strictly positive and the other containing strictly negative solutions. As an application we deduce results on anti-maximum and maximum principles for the p-Laplacian operator with nonlinear boundary conditions.

A Fourier-based library of unbounded Poisson solvers (FLUPS) for 2D and 3D homogeneous distributed grids is presented. It is designed to handle every possible combination of periodic, symmetric, semi-unbounded, and fully unbounded boundary conditions for the Poisson equation on rectangular domains with uniform resolution. FLUPS leverages a dedicated implementation of 3D Fourier transforms to solve the Poisson equation using Green's functions in a fast and memory-efficient way. Several Green's functions are available, optionally with explicit regularization, spectral truncation, or using lattice Green's functions, and provide verified convergence orders from 2 to spectral-like. The algorithm depends on the FFTW library to perform 1D transforms, while message passing interface (MPI) communications enable the required remapping of data in memory. For the latter operation, a first available implementation resorts to the standard all-to-all routines. A second implementation, featuring non-blocking and persistent point-to-point communications, is however shown to be more efficient in a majority of cases and especially while taking advantage of the shared memory parallelism with OpenMP. The scalability of the algorithm, aimed at massively parallel architectures, is demonstrated up to 73720 cores. The results obtained with three different supercomputers show that the weak efficiency remains above 40\% and the strong efficiency above 30\% when the number of cores is multiplied by 16, for typical problems. These figures are slightly better than those expected from a third party 3D fast Fourier transform (FFT) tool, with which a 20\% longer execution time was also measured on average. From the outside, the solving procedure is fully automated so that the user benefits from the optimal performances while not having to handle the complexity associated with memory management, data mapping, and Fourier transform computation. The parallel code is available under Apache license 2.0 at github.com/vortexlab-uclouvain/flups. [ABSTRACT FROM AUTHOR]

We prove an existence result of entropy solution for a class of nonlinear elliptic problems of Leray-Lions type with large monotonicity condition in the framework of weighted Orlicz-Sobolev spaces and with right hand side f ∈ L ¹ (Ω). [ABSTRACT FROM AUTHOR]

Let Ω ⊂ R N ( N ≥ 1 ) be a smooth bounded domain, a ∈ C (Ω ¯) a sign-changing function, and 0 ≤ q < 1 . We investigate the Robin problem where α ∈ [ - ∞ , ∞) and ν is the unit outward normal to ∂ Ω . Due to the lack of strong maximum principle structure, this problem may have dead core solutions. However, for a large class of weights a we recover a positivity property when q is close to 1, which considerably simplifies the structure of the solution set. Such property, combined with a bifurcation analysis and a suitable change of variables, enables us to show the following exactness result for these values of q: (P α) has exactly one nontrivial solution for α ≤ 0 , exactly two nontrivial solutions for α > 0 small, and no such solution for α > 0 large. Assuming some further conditions on a, we show that these solutions lie in a subcontinuum. These results partially rely on (and extend) our previous work (Kaufmann et al. in J Differ Equ 263:4481–4502, 2017), where the cases α = - ∞ (Dirichlet) and α = 0 (Neumann) have been considered. We also obtain some results for arbitrary q ∈ 0 , 1 . Our approach combines mainly bifurcation techniques, the sub-supersolutions method, and a priori lower and upper bounds. [ABSTRACT FROM AUTHOR]

We design an abstract setting for the approximation in Banach spaces of operators acting in duality. A typical example are the gradient and divergence operators in Lebesgue-Sobolev spaces on a bounded domain. We apply this abstract setting to the numerical approximation of Leray-Lions type problems, which include in particular linear diffusion. The main interest of the abstract setting is to provide a unified convergence analysis that simultaneously covers (i) all usual boundary conditions, (ii) several approximation methods. The considered approximations can be conforming (that is, the approximation functions can belong to the energy space relative to the problem) or not, and include classical as well as recent numerical schemes. Convergence results and error estimates are given. We finally briefly show how the abstract setting can also be applied to some models such as flows in fractured medium, elasticity equations and diffusion equations on manifolds. [ABSTRACT FROM AUTHOR]

We deal with the numerical solution of elliptic not necessarily self-adjoint problems. We derive a posteriori upper bound based on the flux reconstruction that can be directly and cheaply evaluated from the original fluxes and we show for one-dimensional problems that local efficiency of the resulting a posteriori error estimators depends on p1/2 only, where p is the discretization polynomial degree. The theoretical results are verified by numerical experiments. [ABSTRACT FROM AUTHOR]

A finite element method (FEM) is proposed to solve second elliptic problems posted in domains with rough boundaries. We also present an FEM for eDiptic problems as an extension of the original method. The optimal error estimates in the energy norm are also proved. Numerical results for second elliptic equations in regular and rough domains are presented to illustrate the efficiency of our method, respectively, which is consistent with theoretical analysis. [ABSTRACT FROM AUTHOR]

In this work, we prove an existence theorem of renormalized solutions for nonlinear elliptic problem of the type - div a (x , u , ∇ u) - div Φ (x , u) = f in Ω , where the lower order term Φ verifies the natural growth condition | Φ (x , s) | ≤ γ (x) + M ¯ - 1 (M (| s |)) , with γ ∈ E M ¯ (Ω). No Δ 2 -condition is needed neither on the N-function M nor on its complementary M ¯ . [ABSTRACT FROM AUTHOR]