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

Optimal control problems are a class of optimisation problems with partial differential equations as constraints. These problems arise in many application areas of science and engineering. The finite element method was used to transform the optimal control problems of an elliptic partial differential equation into a system of linear equations of saddle point form. The main focus of this paper is to characterise and exploit the structure of the coeficient matrix of the saddle point system to build an eficient numerical process. These systems are of large dimension, block, sparse, indefinite and ill conditioned. The numerical solution of saddle point problems is a computational task since well known numerical schemes perform poorly if they are not properly preconditioned. The main task of this paper is to construct a preconditioner the mimic the structure of the system coeficient matrix to accelerate the convergence of the generalised minimal residual method. Explicit expression of the eigenvalue and eigenvectors for the preconditioned matrix are derived. The main outcome is to achieve optimal convergence results in a small number of iterations with respect to the decreasing mesh size h and the changes in fi the regularisation problem parameters. The numerical results demonstrate the effectiveness and performance of the proposed preconditioner compared to the other existing preconditioners and confirm theoretical results. [ABSTRACT FROM AUTHOR]

In uncertainty quantification, the quantity of interest is usually the statistics of the space and/or time integration of system solution. In order to reduce the computational cost, a Bayes estimator based on multilevel Monte Carlo (MLMC) is introduced in this paper. The cumulative distribution function of the output of interest, that is, the expectation of the indicator function, is estimated by MLMC method instead of the classic Monte Carlo simulation. Then, combined with the corresponding probability density function, the quantity of interest is obtained by using some specific quadrature rules. In addition, the smoothing of indicator function and Latin hypercube sampling are used to accelerate the reduction of variance. An elliptic stochastic partial differential equation is used to provide a research context for this model. Numerical experiments are performed to verify the advantage of computational reduction and accuracy improvement of our MLMC-Bayes method. [ABSTRACT FROM AUTHOR]

Here and hereafter, we identify HT ht with a 1-form (not relabelled). Thus HT ht is constant. Nomenclature Throughout HT ht is a simply-connected bounded domain with outward unit normal vectorfield HT ht . Therefore, by Eq. (3), HT ht equals zero. [Extracted from the article]

Let Σ be a C³ compact symmetric convex hypersurface in ℝ8. We prove that when Σ carries exactly four geometrically distinct closed characteristics, then there are at least two irrationally elliptic closed characteristics on Σ. [ABSTRACT FROM AUTHOR]

This work is concerned with second-order necessary and sufficient optimality conditions for optimal control of a non-smooth semilinear elliptic partial differential equation, where the nonlinearity is the non-smooth max-function and thus the associated control-to-state operator is, in general, not Gâteaux-differentiable. In addition to standing assumptions, two main hypotheses are imposed. The first one is the Gâteaux-differentiability at the considered control of the objective functional and it is precisely characterized by the vanishing of an adjoint state on the set of all zeros of the corresponding state. The second one is a structural assumption on the sets of all points at which the values of the interested state are 'close' to the non-differentiability point of the max-function. We then derive a 'no-gap' theory of second-order optimality conditions in terms of a second-order generalized derivative of the cost functional, i.e. for which the only change between necessary and sufficient second-order optimality conditions are between a strict and non-strict inequality. [ABSTRACT FROM AUTHOR]

This paper proposes a novel computational analysis to solve fractional differential equations based on elliptic discrete equations. The traditional solution method of the fractional differential equation was complex, and the solution efficiency and accuracy were low. In the process of solving, the definition, properties, and integral transformation of fractional calculus were analyzed, and then solving the fractional differential equation on the basis of the elliptic discrete equation. From the evaluation of the results, it can be concluded that the proposed method has high solution accuracy. [ABSTRACT FROM AUTHOR]

General elliptic equations with spatially discontinuous diffusion coefficients may be used as a simplified model for subsurface flow in heterogeneous or fractured porous media. In such a model, data sparsity and measurement errors are often taken into account by a randomization of the diffusion coefficient of the elliptic equation which reveals the necessity of the construction of flexible, spatially discontinuous random fields. Subordinated Gaussian random fields are random functions on higher dimensional parameter domains with discontinuous sample paths and great distributional flexibility. In the present work, we consider a random elliptic partial differential equation (PDE) where the discontinuous subordinated Gaussian random fields occur in the diffusion coefficient. Problem specific multilevel Monte Carlo (MLMC) Finite Element methods are constructed to approximate the mean of the solution to the random elliptic PDE. We prove a-priori convergence of a standard MLMC estimator and a modified MLMC—control variate estimator and validate our results in various numerical examples. [ABSTRACT FROM AUTHOR]

In this paper, based on the combination of the Jacobi elliptic equation and the concept of the simple equation method, we introduce a new approach for solving fractional partial differential equations, where the fractional derivative is defined in the sense of the conformable fractional derivative. By use of a nonlinear transformation, the proved chain rule and the properties of fractional calculus, certain fractional partial differential equation can be converted into another ordinary differential equation of integer order. With general solutions of the Jacobi elliptic equation, a series of exact solutions for the ordinary differential equation can be obtained subsequently based on the homogeneous balance principle and with the aid of mathematical software. As for applications of this approach, we apply it to seek exact solutions for the space fractional (2+1)- dimensional breaking soliton equations and the space-time fractional BBM equation. As a result, abundant solitary wave solutions, periodic wave solutions, rational function solutions and Jacobi elliptic function solutions are successfully found. [ABSTRACT FROM AUTHOR]

This paper develops a two-scale asymptotic homogenization method for periodic composite Kirchhoff plates. In this method, a three-dimensional (3D) periodic plate problem is simplified as a Kirchhoff plate problem, which is governed by a fourth-order uniformly elliptic partial differential equation (PDE) with periodically oscillating coefficients. Then, a two-scale solution in an asymptotic expansion form is presented for the PDE, and it is found that the first-order perturbed displacement in the asymptotic solution is zero. Additionally, periodic boundary and normalization constraint conditions are proposed to determine the unique solution to unit cell problems. Moreover, standard finite element formulations for calculating the perturbed displacements are derived from the principle of virtual work. Physical interpretations of the influence functions are presented by analyzing the properties of self-balanced quasi-load vectors used for solving the influence functions. Numerical comparisons show that the present method is physically acceptable and highly accurate. [ABSTRACT FROM AUTHOR]

We present a direct numerical method for the solution of an optimal control problem controlling the growth of LDL (Low-density Lipoprotein), HDL (High-density Lipoprotein) and plaque. The optimal control problem is constrained with a system of coupled nonlinear free and mixed boundary partial differential equations consisting of three parabolics one elliptic and one ordinary differential equations. In the first step, the original problem is transformed from a free boundary problem into a fixed one and from the mixed boundary condition to a Neumann one. Then, employing a fixed point-collocation method, we solve the optimal control problem. In each step of the fixed point iteration, the problem is changed to a linear one and then, the equations are solved using the collocation method bringing about an NLP which is solved using sequential quadratic programming. Then, the obtained solution is verified using indirect methods originating from the first-order optimality conditions. Numerical results are considered to illustrate the efficiency of methods. [ABSTRACT FROM AUTHOR]

In this article, we establish a cluster‐based gradient method (CGM) by combining K‐means clustering algorithm and stochastic gradient descent (SGD). By clustering the sampling solutions, we use the cluster centroids to represent sampling data and give an estimate to the full gradient. It is well known that the full gradient descent (FGD) can provide the steepest descent direction for finding a local minimum of the desired stochastic control problems. However, the huge computational requirements, which is proportional to the product of sample size and the numerical cost for each sample, often makes FGD cost prohibitive for large scale optimization problems. To reduce the formidable cost and the risks of getting stuck in a local minimum, SGD is proposed and can be regarded as a stochastic approximation of FGD. This, however, would result in a slow convergence due to the incorrect approximation of the iteratively update parameters. Our study shows that CGM could provide a good stochastic approximation to the full gradient with small sample size while has a more stable and faster convergence than SGD. To verify our algorithm, a stochastic elliptic control problem is selected and tested. The numerical results validate our method as a reliable gradient descent method with great potential applications in optimization problems. [ABSTRACT FROM AUTHOR]

We consider the case of an anisotropic elastic elliptical inhomogeneity embedded in an infinite anisotropic elastic matrix subjected to a non-uniform remote loading described by remote stresses and strains which are linear functions of the two in-plane coordinates. The internal stresses and strains within the elliptical inhomogeneity are found to be linear functions of the two in-plane coordinates. In addition, we obtain explicit real-form solutions describing the elastic field inside the inhomogeneity as well as hoop stress vectors and hoop stresses on the matrix side and on the inhomogeneity side. We also obtain the corresponding solutions for the two limiting cases in which the elliptical inhomogeneity takes the form of a hole or a rigid inhomogeneity. The solution method presented here can be extended to accommodate the more general scenario in which the remote applied stresses and strains are arbitrary-order polynomials of the two in-plane coordinates. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

Unstructured hex meshes are partitions of three spaces into boxes that can include irregular edges, where n ≠ 4 boxes meet along an edge, and irregular points, where the box arrangement is not consistent with a tensor-product grid. A new class of tri-cubic C 1 splines is evaluated as a tool for solving elliptic higher-order partial differential equations over unstructured hex meshes. Convergence rates for four levels of refinement are computed for an implementation of the isogeometric Galerkin approach applied to Poisson's equation and the biharmonic equation. The ratios of error are contrasted and superior to an implementation of Catmull-Clark solids. For the trivariate Poisson problem on irregularly partitioned domains, the reduction by 2 4 in the L 2 norm is consistent with the optimal convergence on a regular grid, whereas the convergence rate for Catmull-Clark solids is measured as O (h 3 ). The tri-cubic splines in the isogeometric framework correctly solve the trivariate biharmonic equation, but the convergence rate in the irregular case is lower than O( h 4 ). An optimal reduction of 2 4 is observed when the functions on the C 1 geometry are relaxed to be C 0 . [ABSTRACT FROM AUTHOR]

Recently, a multivariate multifractal analysis for pointwise regularities based on hierarchical multiresolution quantities was developed. General bounds between the Hausdorff dimension of the intersection of single fractal sets and that of the original sets were derived. Equalities were checked for some synthetic signals that include multiplicative cascades. In this paper, we focus on the setting supplied by simultaneous pointwise (T u i p i ) i = 1 , ... , L regularities. The T u p regularity for 1 ≤ p < ∞ was first introduced in order to better study elliptic partial differential equations where the natural function space setting is L p or a Sobolev space which includes unbounded functions. We will prove that both corresponding multivariate multifractal formalism and equalities above hold Baire generically in a given product of Besov spaces ∏ i = 1 , ... , L B t i s i , r i (R d) , s i , t i , r i > 0 , 1 ≤ p i ≤ ∞ such that s i − d t i > − d p i . We therefore extend previous results where only cases ( p i = ∞ and s i − d t i > 0 for all i) and ( p i = ∞ , s i = d t i and r i ≤ 1 for all i) for simultaneous pointwise Hölder regularities have been proved. [ABSTRACT FROM AUTHOR]

We deal with the problem of parameter estimation in stochastic differential equations (SDEs) in a partially observed framework. We aim to design a method working for both elliptic and hypoelliptic SDEs, the latters being characterized by degenerate diffusion coefficients. This feature often causes the failure of constrast estimator based on Euler Maruyama discretization scheme and dramatically impairs classic stochastic filtering methods used to reconstruct the unobserved states. All of theses issues make the estimation problem in hypoelliptic SDEs difficult to solve. To overcome this, we construct a well-defined cost function no matter the elliptic nature of the SDEs. We also bypass the filtering step by considering a control theory perspective. The unobserved states are estimated by solving deterministic optimal control problems using numerical methods which do not need strong assumptions on the diffusion coefficient conditioning. Numerical simulations made on different partially observed hypoelliptic SDEs reveal our method produces accurate estimate while dramatically reducing the computational price comparing to other estimation procedures. [ABSTRACT FROM AUTHOR]

We propose a regularization method for ensemble Kalman filtering (EnKF) with elliptic observation operators. Commonly used EnKF regularization methods suppress state correlations at long distances. For observations described by elliptic partial differential equations, such as the pressure Poisson equation (PPE) in incompressible fluid flows, distance localization should be used cautiously, as we cannot disentangle slowly decaying physical interactions from spurious long-range correlations. This is particularly true for the PPE, in which distant vortex elements couple nonlinearly to induce pressure. Instead, these inverse problems have a low effective dimension: low-dimensional projections of the observations strongly inform a low-dimensional subspace of the state space. We derive a low-rank factorization of the Kalman gain based on the spectrum of the Jacobian of the observation operator. The identified eigenvectors generalize the source and target modes of the multipole expansion, independently of the underlying spatial distribution of the problem. Given rapid spectral decay, inference can be performed in the low-dimensional subspace spanned by the dominant eigenvectors. This low-rank EnKF is assessed on dynamical systems with Poisson observation operators, where we seek to estimate the positions and strengths of point singularities over time from potential or pressure observations. We also comment on the broader applicability of this approach to elliptic inverse problems outside the context of filtering. [ABSTRACT FROM AUTHOR]

We consider a model mixing sharp and diffuse interface approaches for seawater intrusion phenomenon in confined aquifers. The problem consists of a strongly coupled system of partial differential equations of parabolic and elliptic types. The aim of this work is to introduce and analyze a new mixed formulation, obtained by writing the system into a matrix form and introducing a new variable σ = R(u)∇u, representing the flux tensor of the primal variable u = (h,Φf)T. Here, h stands for the depth of salt/freshwater interface, Φf, the hydraulic head of fresh water, and R(u) a symmetric and positive definite diffusion matrix. We show that the continuous problem as well as its mixed finite element approximation are well posed. We also provide optimal error estimates in L2 −norm for both variables u and its associated flux σ. [ABSTRACT FROM AUTHOR]

We present a mixed finite element method for approximating a fourth-order elliptic partial differential equation (PDE), the Kirchhoff plate equation, on a surface embedded in |${\mathbb {R}}^{3}$|⁠ , with or without boundary. Error estimates are given in mesh-dependent norms that account for the surface approximation and the approximation of the surface PDE. The method is built on the classic Hellan–Herrmann–Johnson method (for flat domains), and convergence is established for |$C^{k+1}$| surfaces, with degree |$k$| (Lagrangian, parametrically curved) approximation of the surface, for any |$k \geqslant 1$|⁠. Mixed boundary conditions are allowed, including clamped, simply-supported and free conditions; if free conditions are present then the surface must be at least |$C^{2,1}$|⁠. The framework uses tools from differential geometry and is directly related to the seminal work of Dziuk, G. (1988) Finite elements for the Beltrami operator on arbitrary surfaces. Partial Differential Equations and Calculus of Variations , vol. 1357 (S. Hildebrandt & R. Leis eds). Berlin, Heidelberg: Springer, pp. 142–155. for approximating the Laplace–Beltrami equation. The analysis here is the first to handle the full surface Hessian operator directly. Numerical examples are given on nontrivial surfaces that demonstrate our convergence estimates. In addition, we show how the surface biharmonic equation can be solved with this method. [ABSTRACT FROM AUTHOR]

26 F. Seuffert, An extension of the Bianchi-Egnell stability estimate to Bakry, Gentil, and Ledoux's generalization of the Sobolev inequality to continuous dimensions, J. Funct. Morrey's inequality, calculus of variations, extremal, Euler-Lagrange, p-Laplacian, functional inequality, p-Laplace, 49J40, 31C45, nonlinear potential theory, stability estimate, 35J62 Keywords: Morrey's inequality; nonlinear potential theory; calculus of variations; extremal; Euler-Lagrange; p-Laplacian; functional inequality; stability estimate; p-Laplace; 35J62; 49J40; 31C45 EN Morrey's inequality nonlinear potential theory calculus of variations extremal Euler-Lagrange p-Laplacian functional inequality stability estimate p-Laplace 35J62 49J40 31C45 863 877 15 10/03/22 20221001 NES 221001 1 Introduction xtremals, Trans. Amer. Math. Soc. 360 (2008), no. 8, 4335-4347. [Extracted from the article]

26 H. Jia, J. Tao, D. Yang, W. Yuan and Y. Zhang, Special John-Nirenberg-Campanato spaces via congruent cubes, submitted. 40 J. Tao, D. Yang and W. Yuan, John-Nirenberg-Campanato spaces, Nonlinear Anal. 189 (2019), Paper No. 111584. Vanishing John-Nirenberg spaces. [Extracted from the article]

The interpolation finite difference method (IFDM) allows free numerical analysis of elliptic partial differential equations over arbitrary domains. Conventionally, in the finite difference method (FDM), the calculation is performed using the second-order accuracy central difference. For engineering problems, second-order accuracy calculations are often sufficient. On the other hand, much research has been carried out to improve the accuracy of numerical calculations. Although there is much research in the FDM field, the development of numerical calculations by the spectral method is decisive in improving the calculation accuracy. Numerical calculations are usually performed by double precision calculations. If double precision calculations ensure 15 significant digits in floating point computing, such numerical calculations will be the ultimate goal to reach. A numerical calculation that does not seem to have an error even though it originally has an error is defined as a virtual error-zero (VE0) calculation. In this paper, we will examine an overall picture of high-accuracy numerical calculation by the IFDM in the numerical calculation of the 1D Poisson equation. It becomes clear that a VE0 calculation is always possible in the numerical calculation method, defined as the compact interpolation finite difference scheme [(m)]. [ABSTRACT FROM AUTHOR]

A general adaptive refinement strategy for solving linear elliptic partial differential equations with random data is proposed and analysed herein. The adaptive strategy extends the a posteriori error estimation framework introduced by Guignard and Nobile [SIAM J. Numer. Anal., 56 (2018), pp. 3121-3143] to cover problems with a nonaffine parametric coefficient dependence. A suboptimal, but nonetheless reliable and convenient implementation of the strategy involves aproximation of the decoupled PDE problems with a common finite element approximation space. Computational results obtained using such a single-level strategy are presented in this paper (part I). Results obtained using a potentially more efficient multilevel approximation strategy, where meshes are individually tailored, will be discussed in part II of this work. The codes used to generate the numerical results are available on GitHub. [ABSTRACT FROM AUTHOR]

As an extension to the well-established stationary elliptic partial differential equation (PDE) with random continuous coefficients we study a time-dependent advection-diffusion problem, where the coefficients may have random spatial discontinuities. In a subsurface flow model, the randomness in a parabolic equation may account for insufficient measurements or uncertain material procurement, while the discontinuities could represent transitions in heterogeneous media. Specifically, a scenario with coupled advection and diffusion coefficients that are modeled as sums of continuous random fields and discontinuous jump components are considered. The respective coefficient functions allow a very flexible modeling, however, they also complicate the analysis and numerical approximation of the corresponding random parabolic PDE. We show that the model problem is indeed well-posed under mild assumptions and show measurability of the pathwise solution. For the numerical approximation we employ a sample-adapted, pathwise discretization scheme based on a finite element approach. This semi-discrete method accounts for the discontinuities in each sample, but leads to stochastic, finite-dimensional approximation spaces. We ensure measurability of the semi-discrete solution, which in turn enables us to derive moments bounds on the mean-squared approximation error. By coupling this semi-discrete approach with suitable coefficient approximation and a stable time stepping, we obtain a fully discrete algorithm to solve the random parabolic PDE. We provide an overall error bound for this scheme and illustrate our results with several numerical experiments. [ABSTRACT FROM AUTHOR]

We consider a mixed boundary value problem for a strongly elliptic differential–difference equation in a bounded domain. The relationship of this problem with a nonlocal mixed boundary value problem for an elliptic differential equation is established. Theorems on the unique solvability of both problems and on the smoothness of their generalized solutions are stated. [ABSTRACT FROM AUTHOR]

Physics-informed neural networks (PINNs) based machine learning is an emerging framework for solving nonlinear differential equations. However, due to the implicit regularity of neural network structure, PINNs can only find the flattest solution in most cases by minimizing the loss functions. In this paper, we combine PINNs with the homotopy continuation method, a classical numerical method to compute isolated roots of polynomial systems, and propose a new deep learning framework, named homotopy physics-informed neural networks (HomPINNs), for solving multiple solutions of nonlinear elliptic differential equations. The implementation of an HomPINN is a homotopy process that is composed of the training of a fully connected neural network, named the starting neural network, and training processes of several PINNs with different tracking parameters. The starting neural network is to approximate a starting function constructed by the trivial solutions, while other PINNs are to minimize the loss functions defined by boundary condition and homotopy functions, varying with different tracking parameters. These training processes are regraded as different steps of a homotopy process, and a PINN is initialized by the well-trained neural network of the previous step, while the first starting neural network is initialized using the default initialization method. Several numerical examples are presented to show the efficiency of our proposed HomPINNs, including reaction-diffusion equations with a heart-shaped domain. [ABSTRACT FROM AUTHOR]

By a probabilistic method, we prove the existence and uniqueness of weak solutions to Neumann problems for a class of semi-linear elliptic partial differential equations with nonlinear singular divergence terms, which can only be understood in distributional sense. This leads to the further study on a new class of infinite horizon backward stochastic differential equations, which involves integrals with respect to a forward–backward martingale and a singular continuous increasing process. [ABSTRACT FROM AUTHOR]

Recently, the two-dimensional elliptic singularly perturbed boundary value problems have received attention. These problems have not been much explored numerically. A highly accurate numerical scheme on different non-uniform meshes is suggested to solve such problems. In particular, the Haar wavelet method on a special type of non-uniform mesh and Shishkin mesh is proposed; because of the use of the block pulse function, it is easy to derive and handle the operator matrices. Also, it has been shown that Shiskin mesh provides better results. The use of the piecewise-uniform Shishkin mesh with the Haar wavelet scheme contains a novelty in itself. Through rigorous analysis, the method is shown as first-order convergent in L 2 -norm. The theoretical results are confirmed by computational results obtained in the maximum norm and L 2 -norm for two test problems. From the comparative results provided in tables, it is worth noting that the Shishkin mesh is an excellent choice to solve these problems instead of a particular type of non-uniform mesh for the proposed scheme. This can be justified because the grid points defined by the Shishkin mesh are adequately distributed in the layer and non-layer regions. [ABSTRACT FROM AUTHOR]

Upon solving a finite discrete reduction in the difference Heun equation, we arrive at an elliptic generalization of the Racah polynomials. We exhibit the three-term recurrence relation and the orthogonality relations for these elliptic Racah polynomials. The well-known q-Racah polynomials of Askey and Wilson are recovered as a trigonometric limit. [ABSTRACT FROM AUTHOR]

The existence theory is developed for solutions of the inhomogeneous linearized field equations for causal variational principles. These equations are formulated weakly with an integral operator which is shown to be bounded and symmetric on a Hilbert space endowed with a suitably adapted weighted L 2 -scalar product. Guided by the procedure in the theory of linear elliptic partial differential equations, we use the spectral calculus to define Sobolev-type Hilbert spaces and invert the linearized field operator as an operator between such function spaces. The uniqueness of the resulting weak solutions is analyzed. Our constructions are illustrated in simple explicit examples. The connection to the causal action principle for static causal fermion systems is explained. [ABSTRACT FROM AUTHOR]

We propose and analyze a posteriori error estimators for an optimal control problem that involves an elliptic partial differential equation as state equation and a control variable that enters the state equation as a coefficient; pointwise constraints on the control variable are considered as well. We consider two different strategies to approximate optimal variables: a fully discrete scheme in which the admissible control set is discretized with piecewise constant functions and a semi-discrete scheme where the admissible control set is not discretized; the latter scheme being based on the so-called variational discretization approach. We design, for each solution technique, an a posteriori error estimator and show, in two- and three-dimensional Lipschitz polygonal/polyhedral domains (not necessarily convex), that the proposed error estimator is reliable and efficient. We design, based on the devised estimators, adaptive strategies that deliver optimal experimental rates of convergence for the performed numerical examples. [ABSTRACT FROM AUTHOR]

This paper is concerned with the Galerkin spectral approximation of an optimal control problem governed by the elliptic partial differential equations (PDEs). Its objective functional depends on the control variable governed by the L 2 L 2 -norm constraint. The optimality conditions for both the optimal control problem and its corresponding spectral approximation problem are given, successively. Thanks to some lemmas and the auxiliary systems, a priori error estimates of the Galerkin spectral approximation problem are established in detail. Moreover, a posteriori error estimates of the spectral approximation problem are also investigated, which include not only H 1 H 1 -norm error for the state and co-state but also L 2 L 2 -norm error for the control, state and costate. Finally, three numerical examples are executed to demonstrate the errors decay exponentially fast. [ABSTRACT FROM AUTHOR]

In this paper, our main purpose is to study the soliton solutions of conformable time-fractional perturbed Radhakrishnan-Kundu-Lakshmanan equation. New soliton solutions have been obtained by the extended (G'=G)-expansion method, first integral method and complete discrimination system for the polynomial method, respectively. The solutions we obtained mainly include hyperbolic function solutions, solitary wave solutions, Jacobi elliptic function solutions, trigonometric function solutions and rational function solutions. Moreover, we draw its three-dimensional graph. [ABSTRACT FROM AUTHOR]

In this paper we introduce a Model Order Reduction (MOR) algorithm based on a sparse grid adaptive refinement, for the approximation of the eigensolutions to parametric problems arising from elliptic partial differential equations. In particular, we are interested in detecting the crossing of the hypersurfaces describing the eigenvalues as a function of the parameters. The a priori matching is followed by an a posteriori verification, driven by a suitably defined error indicator. At a given refinement level, a sparse grid approach is adopted for the construction of the grid of the next level, by using the marking given by the a posteriori indicator. Various numerical tests confirm the good performance of the scheme. [ABSTRACT FROM AUTHOR]

[Display omitted] • A unified FFT framework with coupled gradient damage model and phase-filed method for the failure analysis of composites is proposed. • The problems of localization and interface damage diffusion in the failure analysis of heterogeneous materials have been effectively overcome. • Interphase debonding and matrix cracking failures in unidirectional composites have been captured objectively. A unified FFT framework coupling the Gradient Damage Model (GDM) and the Variational Phase-Field Model (PFM) for failure analysis of composites is proposed. GDM and PFM are used to simulate the failure of the matrix and the interphase, respectively. Based on the similarity of the damage evolution equations (elliptic partial differential equations) of GDM and PFM, a unified iterative solution and numerical implementation based on fast Fourier transform (FFT) are derived, and simultaneously applied to a computational model. In addition, the high computational efficiency of the FFT method is used to solve the mechanical equilibrium problem. The complete form of the PFM is introduced to avoid erroneous damage diffusion at the different phase interfaces of the composites, and a stress decomposition-based PFM is formulated. 2D and 3D computational examples are established to investigate the mesh insensitivity of the damage models used and the erroneous damage diffusion at interfaces. The applicability of the stress decomposition-based PFM at interphase is discussed in the context of single-edge notched specimens under tension and fiber-reinforced composites with interfacial phases under transverse tension. Finally, the performance of the unified computational framework is evaluated using two simple numerical examples, and the mechanical behavior of unidirectional CF/PEEK composites under transverse tensile loading is simulated based on the unified framework. The results are compared with experiments to evaluate the computational accuracy and practical applicability of the proposed framework. [ABSTRACT FROM AUTHOR]

We introduce a fully nonlinear PDE with a differential form, which unifies several important equations in Kähler geometry including Monge-Ampère equations, J -equations, inverse σ k equations, and deformed Hermitian Yang-Mills (dHYM) equations. We pose some natural positivity conditions on Λ, and prove analytical and algebraic criterion for the solvability of the equation. Our results generalize previous works of G. Chen, J. Song, Datar-Pingali and others. As an application, we prove a conjecture of Collins-Jacob-Yau for dHYM equations with small global phase. [ABSTRACT FROM AUTHOR]

We study a free boundary problem modeling multilayer tumor growth with a small time delay τ, representing the time needed for the cell to complete the replication process. The model consists of two elliptic equations which describe the concentration of nutrient and the tumor tissue pressure, respectively, an ordinary differential equation describing the cell location characterizing the time delay and a partial differential equation for the free boundary. In this paper, we establish the well‐posedness of the problem; namely, first, we prove that there exists a unique flat stationary solution (σ∗,p∗,ρ∗,ξ∗) for all μ>0. The stability of this stationary solution should depend on the tumor aggressiveness constant μ. It is also unrealistic to expect the perturbation to be flat. We show that, under non‐flat perturbations, there exists a threshold μ∗>0 such that (σ∗,p∗,ρ∗,ξ∗) is linearly stable if μ<μ∗ and linearly unstable if μ>μ∗. Furthermore, the time delay increases the stationary tumor size. These are interesting results with mathematical and biological implications. [ABSTRACT FROM AUTHOR]

In this paper, we consider the artificial neural networks for solving the elliptic differential equation with boundary layer, in which the gradient of the solution changes sharply near the boundary layer. The solution of the boundary layer problems poses a huge challenge to both traditional numerical methods and artificial neural network methods. By theoretically analyzing the changing rate of the weights of the first hidden layer near the boundary layer, a mapping strategy is added in traditional neural network to improve the convergence of the loss function. Numerical examples are carried out for the 1D and 2D convection–diffusion equation with boundary layer. The results demonstrate that the modified neural networks significantly improve the ability in approximating the solutions with sharp gradient. [ABSTRACT FROM AUTHOR]

This work derives explicit series reversions for the solution of Calderón's problem. The governing elliptic partial differential equation is \nabla \cdot (A\nabla u)=0 in a bounded Lipschitz domain and with a matrix-valued coefficient. The corresponding forward map sends A to a projected version of a local Neumann-to-Dirichlet operator, allowing for the use of partial boundary data and finitely many measurements. It is first shown that the forward map is analytic, and subsequently reversions of its Taylor series up to specified orders lead to a family of numerical methods for solving the inverse problem with increasing accuracy. The convergence of these methods is shown under conditions that ensure the invertibility of the Fréchet derivative of the forward map. The introduced numerical methods are of the same computational complexity as solving the linearised inverse problem. The analogous results are also presented for the smoothened complete electrode model. [ABSTRACT FROM AUTHOR]

We suggest a field extension of the classical elliptic Ruijsenaars-Schneider model. The model is defined in two different ways which lead to the same result. The first one is via the trace of a chain product of L-matrices which allows one to introduce the Hamiltonian of the model and to show that the model is gauge equivalent to a classical elliptic spin chain. In this way, one obtains a lattice field analogue of the Ruijsenaars-Schneider model with continuous time. The second method is based on investigation of general elliptic families of solutions to the 2D Toda equation. We derive equations of motion for their poles, which turn out to be difference equations in space with a lattice spacing η, together with a zero curvature representation for them. We also show that the equations of motion are Hamiltonian. The obtained system of equations can be naturally regarded as a field generalization of the Ruijsenaars-Schneider system. Its lattice version coincides with the model introduced via the first method. The limit η → 0 is shown to give the field extension of the Calogero-Moser model known in the literature. The fully discrete version of this construction is also discussed. [ABSTRACT FROM AUTHOR]

A systematic two-step procedure is proposed for the derivation of full asymptotic expansions of the solution of elliptic partial differential equations set on a domain perforated with a small hole on which a Dirichlet boundary condition is applied. First, an integral representation of the solution is sought, which enables us to exploit the explicit dependence with respect to the small parameter to predict the correct form of a two-scale ansatz. Second, the terms of the ansatz are characterized by the method of compound asymptotic expansions as the solutions of a cascade of successive interior and exterior domains. This allows us to interpret them as high-order correctors, for which error bounds can be proved using variational estimates. The methodology is illustrated on two different problems: we start by revisiting the perforated Poisson problem with Dirichlet boundary conditions on both the hole and the outer boundary, where we highlight how the method enables us to retrieve very naturally the correct form of the ansatz in the most delicate two-dimensional setting. Then, we provide original and complete asymptotic expansions for a perforated cell problem featuring periodic conditions. [ABSTRACT FROM AUTHOR]