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In this article, exact solutions of the Biswas–Arshed equation are obtained using the extended Weierstrass transformation method (EWTM). This method is widely used in solid-state physics, electrodynamics, and mathematical physics, and it yields exact solution functions involving trigonometric, rational trigonometric, Weierstrass elliptic, wave, and rational functions. The process involves expanding the solution functions of an elliptic differential equation into finite series by transforming them into Weierstrass functions. Furthermore, it generates parametric solutions for nonlinear algebraic equation systems, which are particularly useful in mathematical physics. These solutions are derived using the Mathematica package program. To analyze the behavior of these determined solution functions, the article employs separate two- and three-dimensional graphs showing the real and imaginary components, along with contour and density graphs. These visuals aid in comprehending the physical characteristics exhibited by these solution functions. [ABSTRACT FROM AUTHOR]

In this paper we consider optimal control problems where the control variable is a potential and the state equation is an elliptic partial differential equation of Schrödinger type, governed by the Laplace operator. The cost functional involves the solution of the state equation and a penalization term for the control variable. While the existence of an optimal solution simply follows by the direct methods of the calculus of variations, the regularity of the optimal potential is a difficult question and under the general assumptions we consider, no better regularity than the BV one can be expected. This happens in particular for the cases in which a bang-bang solution occurs, where optimal potentials are characteristic functions of a domain. We prove the BV regularity of optimal solutions through a regularity result for PDEs. Some numerical simulations show the behavior of optimal potentials in some particular cases. [ABSTRACT FROM AUTHOR]

In a sequence of papers, the author examined several statistical affinity measures for selecting the coarse degrees of freedom (CDOFs) or coarse nodes (Cnodes) in algebraic multigrid (AMG) for systems of elliptic partial differential equations (PDEs). These measures were applied to a set of relaxed vectors that exposes the problematic error components. Once the CDOFs are determined using any one of these measures, the interpolation operator is constructed in a bootstrap AMG (BAMG) procedure. However, in a recent paper of Kahl and Rottmann, the statistical least angle regression (LARS) method was utilized in the coarsening procedure and shown to be promising in the CDOF selection. This method is generally used in the statistics community to select the most relevant variables in constructing a parsimonious model for a very complicated and high‐dimensional model or data set (i.e., variable selection for a "reduced" model). As pointed out by Kahl and Rottmann, the LARS procedure has the ability to detect group relations between variables, which can be more useful than binary relations that are derived from strength‐of‐connection, or affinity measures, between pairs of variables. Moreover, by using an updated Cholesky factorization approach in the regression computation, the LARS procedure can be performed efficiently even when the original set of variables is large; and due to the LARS formulation itself (i.e., its l1$$ {l}_1 $$‐norm constraint), sparse interpolation operators can be generated. In this article, we extend the LARS coarsening approach to systems of PDEs. Furthermore, we incorporate some modifications to the LARS approach based on the so‐called elastic net and relaxed lasso methods, which are well known and thoroughly analyzed in the statistics community for ameliorating several major issues with LARS as a variable selection procedure. We note that the original LARS coarsening approach may have addressed some of these issues in similar or other ways but due to the limited details provided there, it is difficult to determine the extent of their similarities. Incorporating these modifications (or effecting them in similar ways) leads to improved robustness in the LARS coarsening procedure, and numerical experiments indicate that the changes lead to faster convergence in the multigrid method. Moreover, the relaxed lasso modification permits an indirect BAMG (iBAMG) extension to the interpolation operator. This iBAMG extension applied in an intra‐ or inter‐variable interpolation setting (i.e., nodal‐based coarsening), as well as in variable‐based coarsening, which will not preserve the nodal structure of a finest‐level discretization on the lower levels of the multilevel hierarchy, will be examined. For the variable‐based coarsening, because of the parsimonious feature of LARS, the performance is reasonably good when applied to systems of PDEs albeit at a substantial additional cost over a nodal‐based procedure. [ABSTRACT FROM AUTHOR]

In this paper we propose a weaker version of Penrose's much heeded Strong Cosmic Censorship (SCC) conjecture, asserting inextendability of maximal Cauchy developments by manifolds with Lipschitz continuous Lorentzian metrics and Riemann curvature bounded in Lp. Lipschitz continuity is the threshold regularity for causal structures, while curvature bounds rule out infinite tidal accelerations, arguing for physical significance of this weaker SCC conjecture. The main result of this paper, under the assumption that no extensions exist with higher connection regularity W loc 1 , p , proves in the affirmative this SCC conjecture with bounded curvature for p sufficiently large, (p > 4 to address uniform bounds, p > 2 without uniform bounds). [ABSTRACT FROM AUTHOR]

The fractional differential equation L^\beta u = f posed on a compact metric graph is considered, where \beta >0 and L = \kappa ^2 - \nabla (a\nabla) is a second-order elliptic operator equipped with certain vertex conditions and sufficiently smooth and positive coefficients \kappa,a. We demonstrate the existence of a unique solution for a general class of vertex conditions and derive the regularity of the solution in the specific case of Kirchhoff vertex conditions. These results are extended to the stochastic setting when f is replaced by Gaussian white noise. For the deterministic and stochastic settings under generalized Kirchhoff vertex conditions, we propose a numerical solution based on a finite element approximation combined with a rational approximation of the fractional power L^{-\beta }. For the resulting approximation, the strong error is analyzed in the deterministic case, and the strong mean squared error as well as the L_2(\Gamma \times \Gamma)-error of the covariance function of the solution are analyzed in the stochastic setting. Explicit rates of convergences are derived for all cases. Numerical experiments for {L = \kappa ^2 - \Delta, \kappa >0} are performed to illustrate the results. [ABSTRACT FROM AUTHOR]

We provide sparse estimates for gradients of solutions to divergence form elliptic partial differential equations in terms of the source data. We give a general result of Meyers (or Gehring) type, a result for linear equations with VMO coefficients and a result for linear equations with Dini continuous coefficients. In addition, we provide an abstract theorem conditional on PDE estimates available. The linear results have the full range of weighted estimates with Muckenhoupt weights as a consequence. [ABSTRACT FROM AUTHOR]

In this study, we explore the effectiveness of elliptic partial differential equations (PDEs) in two and three dimensional space based on Helmholtz equation for simulations of acoustic sound in a very complex environment of propagation. For this purpose, we use an advance and robust numerical technique by utilizing the properties of shifted Chebyshev spectral collocation method. This technique is an extension of the traditional Chebyshev polynomials, incorporating a shift in their argument to enhance flexibility across a wider domain, while retaining an extraordinary numerical characteristic such as orthogonality and spectral convergence making them exceptionally effective in finding the approximate solutions. The exponential order of convergence of the proposed approach is shown both through theoretical and numerical approaches. We provide a number of numerical experiments to verify the theoretical results. The spectral convergence has been substantially enhanced by these numerical examples. The exponential order is further validated by numerical error behaviour in both L2 and L∞ norms. [ABSTRACT FROM AUTHOR]

The paper is devoted to the study of the smoothness of generalized solutions of the first boundary-value problem for a strongly elliptic functional differential equation containing orthotropic contraction transformations of the arguments of the unknown function in the leading part. The problem is considered in a disc, the coefficients of the equation are constant. Orthotropic contraction is understood as different contraction in different variables. Conditions for the conservation of smoothness on the boundaries of neighboring subdomains formed by the action of the contraction transformation group onto the disc are found in explicit form for any right-hand side from the Lebesgue space. [ABSTRACT FROM AUTHOR]

In the framework of Clifford analysis, we study a second‐order elliptic (generally nonstrongly elliptic) system of partial differential equations of the form: ν∂x_fϑ∂x_=0$$ {}^{\nu}\kern-0.1em {\partial}_{\underset{\_}{x}}{f}^{\vartheta}\kern-0.1em {\partial}_{\underset{\_}{x}}=0 $$, where ν∂x_$$ {}^{\nu}\kern-0.1em {\partial}_{\underset{\_}{x}} $$ stands for the Dirac operator with respect to a structural set ν$$ \nu $$. The solutions of this system are known as (ν,ϑ)$$ \left(\nu, \vartheta \right) $$‐inframonogenic functions. Our main purpose is to describe necessary and sufficient conditions for the solvability of a ∂¯$$ \overline{\partial} $$‐problem associated with the sandwich operator ν∂x_(·)ϑ∂x_$$ {}^{\nu}\kern-0.1em {\partial}_{\underset{\_}{x}}{\left(\cdotp \right)}^{\vartheta}\kern-0.1em {\partial}_{\underset{\_}{x}} $$. [ABSTRACT FROM AUTHOR]

Algebraic multigrid (AMG) methods are among the most efficient solvers for linear systems of equations and they are widely used for the solution of problems stemming from the discretization of Partial Differential Equations (PDEs). A severe limitation of AMG methods is the dependence on parameters that require to be fine-tuned. In particular, the strong threshold parameter is the most relevant since it stands at the basis of the construction of successively coarser grids needed by the AMG methods. We introduce a novel deep learning algorithm that minimizes the computational cost of the AMG method when used as a finite element solver. We show that our algorithm requires minimal changes to any existing code. The proposed Artificial Neural Network (ANN) tunes the value of the strong threshold parameter by interpreting the sparse matrix of the linear system as a gray scale image and exploiting a pooling operator to transform it into a small multi-channel image. We experimentally prove that the pooling successfully reduces the computational cost of processing a large sparse matrix and preserves the features needed for the regression task at hand. We train the proposed algorithm on a large dataset containing problems with a strongly heterogeneous diffusion coefficient defined in different three-dimensional geometries and discretized with unstructured grids and linear elasticity problems with a strongly heterogeneous Young's modulus. When tested on problems with coefficients or geometries not present in the training dataset, our approach reduces the computational time by up to 30%. [ABSTRACT FROM AUTHOR]

We derive an extension of the sequential homotopy method that allows for the application of inexact solvers for the linear (double) saddle‐point systems arising in the local semismooth Newton method for the homotopy subproblems. For the class of problems that exhibit (after suitable partitioning of the variables) a zero in the off‐diagonal blocks of the Hessian of the Lagrangian, we propose and analyze an efficient, parallelizable, symmetric positive definite preconditioner based on a double Schur complement approach. For discretized optimal control problems with PDE constraints, this structure is often present with the canonical partitioning of the variables in states and controls. We conclude with numerical results for a badly conditioned and highly nonlinear benchmark optimization problem with elliptic partial differential equations and control bounds. The resulting method allows for the parallel solution of large 3D problems. [ABSTRACT FROM AUTHOR]

We introduce a new overlapping domain decomposition method (DDM) to solve fully nonlinear elliptic partial differential equations (PDEs) approximated with monotone schemes. While DDMs have been extensively studied for linear problems, their application to fully nonlinear PDEs remains limited in the literature. To address this gap, we establish a proof of global convergence of these new iterative algorithms using a discrete comparison principle argument. We also provide a specific implementation for the Monge-Ampére equation. Several numerical tests are performed to validate the convergence theorem. These numerical experiments involve examples of varying regularity. Computational experiments show that method is efficient, robust, and requires relatively few iterations to converge. The results reveal great potential for DDM methods to lead to highly efficient and parallelizable solvers for large-scale problems that are computationally intractable using existing solution methods. [ABSTRACT FROM AUTHOR]

This paper proposes novel computational multiscale methods for linear second-order elliptic partial differential equations in nondivergence form with heterogeneous coefficients satisfying a Cordes condition. The construction follows the methodology of localized orthogonal decomposition (LOD) and provides operator-adapted coarse spaces by solving localized cell problems on a fine scale in the spirit of numerical homogenization. The degrees of freedom of the coarse spaces are related to nonconforming and mixed finite element methods for homogeneous problems. The rigorous error analysis of one exemplary approach shows that the favorable properties of the LOD methodology known from divergence-form PDEs, i.e., its applicability and accuracy beyond scale separation and periodicity, remain valid for problems in nondivergence form. [ABSTRACT FROM AUTHOR]

We deal with probabilistic numerical solutions for linear elliptic equations with Neumann boundary conditions in a Lipschitz domain, by using a probabilistic numerical scheme introduced by Milstein and Tretyakov based on new numerical layer methods. [ABSTRACT FROM AUTHOR]

AbstractWe prove the uniformity of the in-plane and anti-plane elastic field of stresses and strains inside an incompressible nonlinear elastic elliptical inhomogeneity embedded in an infinite linear isotropic elastic matrix subjected to uniform remote in-plane and anti-plane stresses. The elastic material occupying the elliptical inhomogeneity obeys Neuber’s special nonlinear stress-strain law. The original boundary value problem is finally reduced to a single non-linear equation for the constant effective strain within the inhomogeneity, which is rigorously proved to have a unique solution. Once the non-linear equation is solved numerically, we establish the uniform elastic field within the elliptical inhomogeneity and the non-uniform elastic field in the linear elastic matrix. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

For the spatial discretization of elliptic and parabolic partial differential equations (PDEs), we provide a Matrix-Oriented formulation of the classical Finite Element Method, called MO-FEM, of arbitrary order k ∈ N. On a quite general class of 2D domains, namely separable domains , and even on special surfaces, the discrete problem is then reformulated as a multiterm Sylvester matrix equation where the additional terms account for the geometric contribution of the domain shape. By considering the IMEX Euler method for the PDE time discretization, we obtain a sequence of these equations. To solve each multiterm Sylvester equation, we apply the matrix-oriented form of the Preconditioned Conjugate Gradient (MO-PCG) method with a matrix-oriented preconditioner that captures the spectral properties of the Sylvester operator. Solving the Poisson problem and the heat equation on some separable domains by MO-FEM-PCG, we show a gain in computational time and memory occupation wrt the classical vector PCG with same preconditioning or wrt a LU based direct method. As an application, we show the advantages of the MO-FEM-PCG to approximate Turing patterns on some separable domains and cylindrical surfaces for a morphochemical reaction-diffusion PDE system for battery modelling. [ABSTRACT FROM AUTHOR]

Spatial network models are used as a simplified discrete representation in a wide range of applications, e.g., flow in blood vessels, elasticity of fiber based materials, and pore network models of porous materials. Nevertheless, the resulting linear systems are typically large and poorly conditioned and their numerical solution is challenging. This paper proposes a numerical homogenization technique for spatial network models which is based on the super-localized orthogonal decomposition (SLOD), recently introduced for elliptic multiscale partial differential equations. It provides accurate coarse solution spaces with approximation properties independent of the smoothness of the material data. A unique selling point of the SLOD is that it constructs an almost local basis of these coarse spaces, requiring less computations on the fine scale and achieving improved sparsity on the coarse scale compared to other state-of-the-art methods. We provide an a posteriori analysis of the proposed method and numerically confirm the method's unique localization properties. In addition, we show its applicability also for high-contrast channeled material data. [ABSTRACT FROM AUTHOR]

The aim of this paper is to study the existence and uniqueness of weak solution for the problem...is bounded and open. Here, the functions f: O R and g: O × R R satisfy the suitable hypotheses. [ABSTRACT FROM AUTHOR]

The theory of mixed finite element methods for solving different types of elliptic partial differential equations in saddle point formulation is well established since many decades. This topic was mostly studied for variational formulations defined upon the same product spaces of both shape- and test-pairs of primal variable-multiplier. Whenever either these spaces or the two bilinear forms involving the multiplier are distinct, the saddle point problem is asymmetric. The three inf-sup conditions to be satisfied by the product spaces stipulated in work on the subject, to guarantee well-posedness, are known [(see, e.g., Exercise 2.14 of Ern and Guermond (Theory and practice of finite elements, Applied mathematical, sciences, Springer, 2004)]. However, the material encountered in the literature addressing the approximation of this class of problems left room for improvement and clarifications. After making a brief review of the existing contributions to the topic that justifies such an assertion, in this paper we set up finer global error bounds for the pair primal variable-multiplier solving an asymmetric saddle point problem. Besides well-posedness, the three constants in the aforementioned inf-sup conditions are identified as all that is needed for determining the stability constant appearing therein, whose expression is exhibited. As a complement, refined error bounds depending only on these three constants are given for both unknowns separately. [ABSTRACT FROM AUTHOR]

This paper is concerned with a regularity analysis of parametric operator equations with a perspective on uncertainty quantification. We study the regularity of mappings between Banach spaces near branches of isolated solutions that are implicitly defined by a residual equation. Under s -Gevrey assumptions on the residual equation, we establish s -Gevrey bounds on the Fréchet derivatives of the locally defined data-to-solution mapping. This abstract framework is illustrated in a proof of regularity bounds for a semilinear elliptic partial differential equation with parametric and random field input. [ABSTRACT FROM AUTHOR]

In this work, we establish existence results for the following fourth-order Kirchhoff-type elliptic problem with Hardy potential M(∫Ω |∆u| pdx) ∆²pu - a |x|p|u|p-2u = λf(x, u), in Ω, u = ∆u = 0, on ∂Ω. Precisely, by using the classical Hardy inequality and critical point theory, we prove the existence of multiple weak solutions for the fourth-order Kirchhoff-type elliptic problem with Hardy potential. [ABSTRACT FROM AUTHOR]

A key observation underlying this paper is the fact that the range invariance condition for convergence of regularization methods for nonlinear ill-posed operator equations—such as coefficient identification in partial differential equations (PDEs) from boundary observations—can often be achieved by extending the searched for parameter in the sense of allowing it to depend on additional variables. This clearly counteracts unique identifiability of the parameter, though. The second key idea of this paper is now to restore the original restricted dependency of the parameter by penalization. This is shown to lead to convergence of variational (Tikhonov type) and iterative (Newton-type) regularization methods. We concretize the abstract convergence analysis in a framework typical of parameter identification in PDEs in a reduced and an all-at-once setting. This is further illustrated by three examples of coefficient identification from boundary observations in elliptic and time-dependent PDEs. [ABSTRACT FROM AUTHOR]

We consider a general nonsymmetric second-order linear elliptic partial differential equation in the framework of the Lax–Milgram lemma. We formulate and analyze an adaptive finite element algorithm with arbitrary polynomial degree that steers the adaptive meshrefinement and the inexact iterative solution of the arising linear systems. More precisely, the iterative solver employs, as an outer loop, the so-called Zarantonello iteration to symmetrize the system and, as an inner loop, a uniformly contractive algebraic solver, for example, an optimally preconditioned conjugate gradient method or an optimal geometric multigrid algorithm. We prove that the proposed inexact adaptive iteratively symmetrized finite element method leads to full linear convergence and, for sufficiently small adaptivity parameters, to optimal convergence rates with respect to the overall computational cost, i.e. the total computational time. Numerical experiments underline the theory. [ABSTRACT FROM AUTHOR]

For any compact connected one-dimensional submanifold |$K\subset \mathbb R^{2\times 2}$| without boundary that has no rank-one connection and is elliptic, we prove the quantitative rigidity estimate $$\begin{align*} \inf_{M\in K}\int_{B_{1/2}}| Du -M |^2\, \textrm{d}x \leq C \int_{B_1} \operatorname{dist}^2(Du, K)\, \textrm{d}x, \qquad\forall u\in H^1(B_1;\mathbb R^2). \end{align*}$$ This is an optimal generalization, for compact connected submanifolds of |$\mathbb R^{2\times 2}$| without boundary, of the celebrated quantitative rigidity estimate of Friesecke, James, and Müller for the approximate differential inclusion into |$SO(n)$|⁠. The proof relies on the special properties of elliptic subsets |$K\subset{{\mathbb{R}}}^{2\times 2}$| with respect to conformal–anticonformal decomposition, which provide a quasilinear elliptic partial differential equation satisfied by solutions of the exact differential inclusion |$Du\in K$|⁠. We also give an example showing that no analogous result can hold true in |$\mathbb R^{n\times n}$| for |$n\geq 3$|⁠. [ABSTRACT FROM AUTHOR]

We describe a compact finite-difference discretization and Gaussian-radial basis function for the two-dimensional local fractional elliptic PDEs that describe anomalous diffusion-convection of groundwater contamination. Precisely estimating pollutant concentration over a long period helps protect water reservoirs. The local fractional partial differential equations and their discretization described here are the generalization of the integer order elliptic partial differential equations and their high-order scheme. The high-order discretization of fractal gradient and anomalous diffusion on a non-uniformly spaced nine-point single-cell grid network gives the result in small computing time. The new scheme is supported by a detailed convergence analysis describing the monotone property and a strongly connected Jacobian (iteration) matrix graph. The computational illustration of various anomalous diffusion-convection models demonstrates the proposed methodology's effectiveness. [ABSTRACT FROM AUTHOR]

For various kinds of parabolic and elliptic partial differential and differential-difference equations, results on the stabilization of solutions are presented. For the Cauchy problem for parabolic equations, the stabilization is treated as the existence of a limit as the time unboundedly increases. For the half-space Dirichlet problem for parabolic equations, the stabilization is treated as the existence of a limit as the independent variable orthogonal to the boundary half-plane unboundedly increases. In the classical case of the heat equation, the necessary and sufficient condition of the stabilization consists of the existence of the limit of mean values of the initial-value (boundary-value) function over balls as the ball radius tends to infinity. For all linear problems considered in the present paper, this property is preserved (including elliptic equations and differential-difference equations). The Wiener Tauberian theorem is used to establish this property. To investigate the differential-difference case, we use the fact that translation operators are Fourier multipliers (as well as differential operators), which allows one to use a standard Gel'fand-Shilov operational scheme. For all quasilinear problems considered in the present paper, the mean value from the stabilization criterion is changed: It undergoes a monotonic map, which is explicitly constructed for each investigated nonlinear boundary-value problem. [ABSTRACT FROM AUTHOR]

In a recent paper, the author examined a correlation affinity measure for selecting the coarse degrees of freedom (CDOFs) or coarse nodes (C nodes) in systems of elliptic partial differential equations (PDEs). This measure was applied to a set of relaxed vectors, which exposed the near‐nullspace components of the PDE operator. Selecting the CDOFs using this affinity measure and constructing the interpolation operators using a least‐squares procedure, an algebraic multigrid (AMG) method was developed. However, there are several noted issues with this AMG solver. First, to capture strong anisotropies, a large number of test vectors may be needed; and second, the solver's performance can be sensitive to the initial set of random test vectors. Both issues reflect the sensitive statistical nature of the measure. In this article, we derive several other statistical measures that ameliorate these issues and lead to better AMG performance. These measures are related to a Markov process, which the PDE itself may model. Specifically, the measures are based on the diffusion distance/effective resistance for such process, and hence, these measures incorporate physics into the CDOF selection. Moreover, because the diffusion distance/effective resistance can be used to analyze graph networks, these measures also provide a very economical scheme for analyzing large‐scale networks. In this article, the derivations of these measures are given, and numerical experiments for analyzing networks and for AMG performance on weighted‐graph Laplacians and systems of elliptic boundary‐value problems are presented. [ABSTRACT FROM AUTHOR]

In an open, bounded Lipschitz polygon Ω ⊂ R 2 , we establish weighted analytic regularity for a semilinear, elliptic PDE with analytic nonlinearity and subject to a source term f which is analytic in Ω . The boundary conditions on each edge of ∂ Ω are either homogeneous Dirichlet or homogeneous Neumann BCs. The presently established weighted analytic regularity of solutions implies exponential convergence of various approximation schemes: hp-finite elements, reduced order models via Kolmogorov n-widths of solution sets in H 1 (Ω) , quantized tensor formats and certain deep neural networks. [ABSTRACT FROM AUTHOR]

The objective of this paper is to establish a connection between the problem of optimal regularity among solutions to elliptic partial differential equations with measurable coefficients and the Liouville property at infinity. Initially, we address the two‐dimensional case by proving an Alt–Caffarelli–Friedman‐type monotonicity formula, enabling the proof of optimal regularity and the Liouville property for multiphase problems. In higher dimensions, we delve into the role of monotonicity formulas in characterizing optimal regularity. By employing a hole‐filling technique, we present a distinct "almost‐monotonicity" formula that implies Hölder regularity of solutions. Finally, we explore the interplay between the least growth at infinity and the exponent of regularity by combining blow‐up and G$G$‐convergence arguments. [ABSTRACT FROM AUTHOR]

In this paper, we give a computationally optimal relaxed scalar auxiliary variable (SAV) approach for solving gradient flow systems, in which we only need to solve an elliptic partial differential equation with constant coefficients at each time step. In addition to being applicable to various types of auxiliary variables, there are several advantages of our methods, including: (i) by modifying the optimization procedure in correction of auxiliary variables into a linear programming problems, the original hard-to-solve nonlinear optimization problem arising from previous relaxed SAV approach introduced in Jiang et al. (2022) [28] can be avoided; (ii) the resulting method yields some novel linear unconditionally energy stable schemes, in which backward Euler and Crank–Nicolson formulas are used to discretize the time so that the accuracy can reach the first- and second-order, respectively; (iii) comparing with the baseline SAV approach, the discrete energy in the energy dissipation law is closer to the original energy. Finally, ample numerical results demonstrate the accuracy, efficiency and energy stability of the proposed scheme. [ABSTRACT FROM AUTHOR]

This paper offers a number of examples showing that in the case of two independent variables the uniform ellipticity of a linear system of differential equations with partial derivatives of the second order, which fulfills condition (3), do not always cause the normal solvability of formulated exterior elliptic problems in the sense of Noether. Nevertheless, from the system of differential equations with partial derivatives of elliptic type it is possible to choose, under certain additional conditions, classes which are normally solvable in the sense of Noether. This paper also shows that for the so-called decomposed system of differential equations, with partial derivatives of an elliptic type in the case of exterior regions, the Noether theorems are valid. [ABSTRACT FROM AUTHOR]

We study the existence of solutions for the nonlinear elliptic problem - Δ ⁢ v - λ ⁢ g 1 ⁢ v + h 1 ⁢ (v) = f 1 in ⁢ V ∖ V 0 , v = 0 on ⁢ V 0 , where V is the Sierpiński gasket in ℝ N - 1 ( N ≥ 2 ), V 0 is its boundary (consisting of its N corners) and λ ∈ ℝ . Here f 1 , g 1 : V → ℝ , h 1 : ℝ → ℝ are the maps satisfying suitable hypotheses. [ABSTRACT FROM AUTHOR]

Sufficient conditions for the existence of a generalized solution to a nonlinear elliptic differential equation with nonlocal boundary conditions of Bitsadze–Samarskii type are proved. The strong ellipticity condition is used for an auxiliary differential-difference operator. Under the formulated conditions, the differential-difference operator is demicontinuous, coercive, and has a semibounded variation, so the general theory of pseudomonotone operators can be applied. [ABSTRACT FROM AUTHOR]

This paper presents a novel multi-scale method for elliptic partial differential equations with arbitrarily rough coefficients. In the spirit of numerical homogenization, the method constructs problem-adapted ansatz spaces with uniform algebraic approximation rates. Localized basis functions with the same super-exponential localization properties as the recently proposed Super-Localized Orthogonal Decomposition enable an efficient implementation. The method's basis stability is enforced using a partition of unity approach. A natural extension to higher order is presented, resulting in higher approximation rates and enhanced localization properties. We perform a rigorous a priori and a posteriori error analysis and confirm our theoretical findings in a series of numerical experiments. In particular, we demonstrate the method's applicability for challenging high-contrast channeled coefficients. [ABSTRACT FROM AUTHOR]

We employ the Craig-Wayne-Bourgain (CWB) method to construct quasi-periodic solutions for the nonlinear delayed perturbation equations. The linearized equation at each step of the iterations is a differential-difference equation on the torus. We shall combine the techniques of Green's function estimate and the reducibility method in KAM theory to solve the linear equation, which generalizes the applicability of the CWB method. As an application, we study the positive quasi-periodic solutions for a class of Lotka-Volterra equations with quasi-periodic coefficients and time delay. [ABSTRACT FROM AUTHOR]

In the current research, we delve into the intricate realm of bifurcation analysis for Föppl–von Kármán plates, employing a precise numerical tool. This innovative numerical approach melds the power of spectral discretization with the prowess of a high-order continuation method-based Taylor series development (HODC). It is worth noting that combining the high-order continuation method with such discretization techniques offers an efficient path-following approach, complete with adaptive step lengths, capable of tackling a wide array of nonlinear problems. Despite the extensive applications of nonlinear elasticity, the spectral method remains relatively uncharted territory within this context. However, our deep-rooted understanding and expertise in the field drive us to embrace this method alongside high-order development continuation for bifurcation analysis of Föppl–von Kármán plates. The governing equations governing thin elastic plates experiencing significant elastic deflections manifest as a pair of coupled nonlinear differential equations, famously known as the von Kármán (vK) equations, presented in a strong form with two principal unknowns: deflection (w) and the Airy stress function (F). Leveraging Chebyshev decomposition matrices, we approximate these fourth-order elliptic nonlinear partial differential equations. Subsequently, we harness high-order development continuation techniques to morph these nonlinear systems into linear ones. Our rigorous evaluation and validation of this numerical approach's precision and performance come to fruition through a comprehensive buckling analysis encompassing multiple illustrative examples. [ABSTRACT FROM AUTHOR]

The sample average approximation (SAA) approach is applied to risk-neutral optimization problems governed by semilinear elliptic partial differential equations with random inputs. After constructing a compact set that contains the SAA critical points, we derive nonasymptotic sample size estimates for SAA critical points using the covering number approach. Thereby, we derive upper bounds on the number of samples needed to obtain accurate critical points of the risk-neutral PDE-constrained optimization problem through SAA critical points. We quantify accuracy using expectation and exponential tail bounds. Numerical illustrations are presented. [ABSTRACT FROM AUTHOR]

In the present paper, we study the analyticity of the leftmost eigenvalue of the linear elliptic partial differential operators with random coefficient and analyse the convergence rate of the quasi-Monte Carlo method for approximation of the expectation of this quantity. The random coefficient is assumed to be represented by an affine expansion $ a_0({\boldsymbol {x}})+\sum _{j\in {\mathbb N}}y_ja_j({\boldsymbol {x}}) $ a 0 (x) + ∑ j ∈ N y j a j (x) , where elements of the parameter vector $ {\boldsymbol {y}}=(y_j)_{j\in {\mathbb N}}\in U^\infty $ y = (y j) j ∈ N ∈ U ∞ are independent and identically uniformly distributed on $ U:=[-\frac {1}{2},\frac {1}{2}] $ U := [ − 1 2 , 1 2 ]. Under the assumption $ \|\sum _{j\in {\mathbb N}}\rho _j|a_j|\|_{L_\infty (D)} ‖ ∑ j ∈ N ρ j | a j | ‖ L ∞ (D) < ∞ with some positive sequence $ (\rho _j)_{j\in {\mathbb N}}\in \ell _p({\mathbb N}) $ (ρ j) j ∈ N ∈ ℓ p (N) for $ p\in (0,1] $ p ∈ (0 , 1 ] we show that for any $ {\boldsymbol {y}}\in U^\infty $ y ∈ U ∞ , the elliptic partial differential operator has a countably infinite number of eigenvalues $ (\lambda _j({\boldsymbol {y}}))_{j\in {\mathbb N}} $ (λ j (y)) j ∈ N which can be ordered non-decreasingly. Moreover, the spectral gap $ \lambda _2({\boldsymbol {y}})-\lambda _1({\boldsymbol {y}}) $ λ 2 (y) − λ 1 (y) is uniformly positive in $ U^\infty $ U ∞ . From this, we prove the holomorphic extension property of $ \lambda _1({\boldsymbol {y}}) $ λ 1 (y) to a complex domain in $ {\mathbb C}^\infty $ C ∞ and estimate partial derivatives of $ \lambda _1({\boldsymbol {y}}) $ λ 1 (y) with respect to the parameter $ {\boldsymbol {y}} $ y by using Cauchy's formula for analytic functions. Based on these bounds we prove the dimension-independent convergence rate of the quasi-Monte Carlo method to approximate the expectation of $ \lambda _1({\boldsymbol {y}}) $ λ 1 (y). [ABSTRACT FROM AUTHOR]

In this paper, a new extension of the perturbed nonlinear Chen-Lee-Liu equation as a variable coefficients model represents optical pulse propagation in a monomode fiber is studied. The direct similarity reduction method is used to transform the Chen-Lee-Liu model with variable coefficients to a nonlinear ordinary differential equation and the Jacobi elliptic expansion method is used to solve the reduced equation and as a result, novel optical solitons, periodic, and singular waves have arisen. Then, to prove the physical existence and importance of the presented variable coefficients Chen-Lee-Liu model, another similarity technique is used to transform the Chen-Lee-Liu model to the generalized derivative Schrödinger equation which has known bright and dark soliton solutions. Finally, a graphical representation of the obtained wave solutions according to different structures of the variable coefficients is presented. [ABSTRACT FROM AUTHOR]

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In this paper, we consider, from a numerical point of view, a two-temperature poro-thermoelastic problem. The model is written as a coupled linear system of hyperbolic and elliptic partial differential equations. An existence result is proved and energy decay properties are recalled. Then we introduce a fully discrete approximation by using the finite element method and the implicit Euler scheme. Some a priori error estimates are obtained, from which the linear convergence of the approximation is deduced under an appropriate additional regularity. Finally, some numerical simulations are performed to demonstrate the accuracy of the approximation, the decay of the discrete energy and the behaviour of the solution depending on a constitutive parameter. [ABSTRACT FROM AUTHOR]

This paper offers a number of examples showing that in the case of two independent variables the uniform ellipticity of a linear system of differential equations with partial derivatives of the second order, which fulfills condition (3), do not always cause the normal solvability of formulated exterior elliptic problems in the sense of Noether. Nevertheless, from the system of differential equations with partial derivatives of elliptic type it is possible to choose, under certain additional conditions, classes which are normally solvable in the sense of Noether. This paper also shows that for the so-called decomposed system of differential equations, with partial derivatives of an elliptic type in the case of exterior regions, the Noether theorems are valid. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

This paper deals with a high-resolution algorithm that engages fuzzy transform to solve three-dimensional nonlinear elliptic partial differential equations. The scheme approximates the fuzzy components, which estimate fourth-order accurate solutions at the interior mesh points of the solution domain. The fuzzy components and triangular base functions will be approximated with a nineteen-point linear combination of solution values and related to exact solutions by a linear system. Such an arrangement along with compact discretization yields a block tridiagonal Jacobian matrix, and an iterative solver can efficiently compute them. The convergence analysis and error bound of the scheme are examined in detail. The method provides an order-preserving solution and applies to a comprehensive class of partial differential equations with nonlinear first-order partial derivatives. Numerical simulations with Helmholtz equation, advection–diffusion–reaction equation, and nonlinear elliptic sine–Gordan equation corroborate the utility, convergence rate, and enhance solution accuracy by employing a new scheme. [ABSTRACT FROM AUTHOR]