Linear Elliptic Boundary Value Problems Research Articles

Positive Jacobian constraints for elliptic boundary value problems with piecewise-regular coefficients arising from multi-wave inverse problems

Multi-wave inverse problems are indirect imaging methods using the interaction of two different imaging modalities. One brings spatial accuracy, and the other contrast sensitivity. The inversion method typically involve two steps. The first step is devoted to accessing internal datum of quantities related to the unknown parameters being observed. The second step involves recovering the parameters themselves from the internal data. To perform that inversion, a typical requirement is that the Jacobian of fields involved does not vanish. A number of authors have considered this problem in the past two decades, and a variety of methods have been developed. Existing techniques require Hölder continuity of the parameters to be reconstructed. In practical applications, the medium may present embedded elements, with distinct physical properties, leading to discontinuous coefficients. In this article we explain how a Jacobian constraint can be imposed in the piecewise regular case, when the physical model is a divergence form second order linear elliptic boundary value problem.

Adaptive multigrid method for quantum eigenvalue problems

In this paper, a new type of adaptive finite element method is proposed for nonlinear eigenvalue problems in electronic structure calculations based on the multilevel correction method and adaptive multigrid method. Different from the classical adaptive finite element method for electronic structure calculations which needs to solve a nonlinear eigenvalue model directly in each adaptive finite element space, our approach only needs to solve a linear elliptic boundary value problem by adaptive multigrid method in each adaptive space, and then correct the eigenpair approximation by solving a small-scale nonlinear eigenvalue model in a low-dimensional correction space. Since solving large-scale nonlinear eigenvalue model is avoided which has exponentially increased computational time, the efficiency of electronic structure calculations can be improved evidently. In addition, the corresponding convergence analysis of the proposed adaptive multigrid algorithm are also derived theoretically and numerically.

Exponential convergence of hp FEM for spectral fractional diffusion in polygons

For the spectral fractional diffusion operator of order 2s, s in (0,1), in bounded, curvilinear polygonal domains varOmega subset {{mathbb {R}}}^2 we prove exponential convergence of two classes of hp discretizations under the assumption of analytic data (coefficients and source terms, without any boundary compatibility), in the natural fractional Sobolev norm {mathbb {H}}^s(varOmega ). The first hp discretization is based on writing the solution as a co-normal derivative of a 2+1-dimensional local, linear elliptic boundary value problem, to which an hp-FE discretization is applied. A diagonalization in the extended variable reduces the numerical approximation of the inverse of the spectral fractional diffusion operator to the numerical approximation of a system of local, decoupled, second order reaction-diffusion equations invarOmega . Leveraging results on robust exponential convergence of hp-FEM for second order, linear reaction diffusion boundary value problems in varOmega , exponential convergence rates for solutions uin {mathbb {H}}^s(varOmega ) of mathcal {L}^s u = f follow. Key ingredient in this hp-FEM are boundary fitted meshes with geometric mesh refinement towardspartial varOmega . The second discretization is based on exponentially convergent numerical sinc quadrature approximations of the Balakrishnan integral representation of mathcal {L}^{-s} combined with hp-FE discretizations of a decoupled system of local, linear, singularly perturbed reaction-diffusion equations invarOmega . The present analysis for either approach extends to (polygonal subsets {widetilde{mathcal {M}}} of) analytic, compact 2-manifolds {mathcal {M}}, parametrized by a global, analytic chart chi with polygonal Euclidean parameter domain varOmega subset {{mathbb {R}}}^2. Numerical experiments for model problems in nonconvex polygonal domains and with incompatible data confirm the theoretical results. Exponentially small bounds on Kolmogorov n-widths of solution sets for spectral fractional diffusion in curvilinear polygons and for analytic source terms are deduced.

A novel adaptive finite element method for the ground state solution of Bose-Einstein condensates

Abstract In this study we propose a novel adaptive finite element method for the ground state solution of Bose-Einstein Condensates (BEC). Different from the classical adaptive scheme applied to BEC which needs to solve a nonlinear eigenvalue model directly on each adaptive finite element space, our scheme requires to solve a linear elliptic boundary value problem on current adaptive space and a nonlinear eigenvalue model on a quite low-dimensional correction space. Further, the linear elliptic boundary value problem is solved by adaptive multigrid iterations. Since there is no nonlinear eigenvalue model to be solved directly on the adaptive spaces, the solving efficiency can be improved evidently. In addition, the convergence analysis of the proposed adaptive algorithm are derived numerically and theoretically.

Estimates for solutions of a class of linear elliptic degenerate operators in Besov-type spaces

In this paper, we give a priori estimates in Besov-type spaces for solutions of a class of linear elliptic degenerate boundary value problems of Baouendi–Grushin type. This work extends some well known results proved in classical Sobolev, Holder and Besov spaces.

Newtonian potential and positive solutions of Poisson equations

We study the properties of the well-known Newtonian potential operator including its domain, codomain, compactness, relation with the Poisson’s equation and the spectral radii of the weight Newtonian potential operators. These new properties are then applied to study the existence of positive classical solutions of nonlinear Poisson’s equations in bounded open sets, which are required neither to be connected nor smooth on their boundaries. The connectedness and smoothness are common requirements for studying the existence of classical or weak solutions of linear or nonlinear elliptic boundary value problems in the literature. Some applications to population models are provided.

Wavelet kernel function based multiscale LSSVM for elliptic boundary value problems

Recently, Mehrkanoon and Suykens designed a least square support vector machine (LSSVM) for learning solutions to partial differential equations in [1], where the Gaussian radial basis function is used as the kernel of the LSSVM. The purpose of the present paper is twofold: firstly, we extend the Gaussian kernel to wavelet kernel; and secondly, we propose a multiscale scheme, by noticing the multiscale nature of the wavelet kernel functions. The multiscale algorithm consists of a sequence of residual corrections to the solution of the partial differential equation, in which different scale parameters are employed to accommodate information at different scales. However, the direct computation of the multiscale problem would be difficult due to the mixing of different scales. To resolve this problem, we further introduce a multilevel algorithm, which decompose the multiscale algorithm into multiple levels: on the first level, a coarse data set and a large scale parameter are chosen and the target function is interpolated in this data set to capture the large-scale variations of the target function; next, on the second level, a smaller scale parameter is used to interpolate the residuals on a finer data set, capturing the finer details. The sum of both interpolants obviously better approximates the target function at the data sites on the finer data set. This process can be further applied to finer and finer scales till the anticipated accuracy is achieved. Note that sometimes the correction on the finer scale is necessary only at some local sites, thus an adaptive algorithm with only local corrections on finer scale is introduced. The numerical tests on some linear second order elliptic boundary value problems show the efficiency of the multilevel algorithm and the adaptive algorithm.

Existence and multiplicity results for a semilinear elliptic problem

In this paper we establish sufficient conditions for an asymptotically linear elliptic boundary value problem to have at least five solutions when the range of the derivative of the nonlinearity includes at least the first two eigenvalues. We make extensive use of variational methods and characterizations of the local degree of critical points. Furthermore, uniqueness and qualitative properties of the solutions are investigated.

An Augmented IB Method & Analysis for Elliptic BVP on Irregular Domains

The immersed boundary method is well-known, popular, and has had vast areas of applications due to its simplicity and robustness even though it is only first order accurate near the interface. In this paper, an immersed boundary-augmented method has been developed for linear elliptic boundary value problems on arbitrary domains (exterior or interior) with a Dirichlet boundary condition. The new method inherits the simplicity, robustness, and first order convergence of the IB method but also provides asymptotic first order convergence of partial derivatives. Numerical examples are provided to confirm the analysis.

Eigencurves for linear elliptic equations

This paper provides results forvariational eigencurvesassociated with self-adjoint linear elliptic boundary value problems. The elliptic problems are treated as a general two-parameter eigenproblem for a triple (a,b,m) of continuous symmetric bilinear forms on a real separable Hilbert spaceV.Geometric characterizationsof eigencurves associated with (a,b,m) are obtained and are based on their variational characterizations described here. Continuity, differentiability, as well as asymptotic, results for these eigencurves are proved. Finally, two-parameter Robin–Steklov eigenproblems are treated to illustrate the theory.

Gradient Weighted Norm Inequalities for Linear Elliptic Equations with Discontinuous Coefficients

Local and global weighted norm estimates involving Muckenhoupt weights are obtained for gradient of solutions to linear elliptic Dirichlet boundary value problems in divergence form over a Lipschitz domain $$\Omega $$ . The gradient estimates are obtained in weighted Lebesgue and Lorentz spaces, which also yield estimates in Lorentz–Morrey spaces as well as Holder continuity of solutions. The significance of the work lies on its applicability to very weak solutions (that belong to $$W^{1,p}_{0}(\Omega )$$ for some $$p>1$$ but not necessarily in $$W^{1,2}_{0}(\Omega )$$ ) to inhomogeneous equations with coefficients that may have discontinuities but have a small mean oscillation. The domain is assumed to have a Lipschitz boundary with small Lipschitz constant and as such allows corners. The approach implemented makes use of localized sharp maximal function estimates as well as known regularity estimates for very weak solutions to the associated homogeneous equations. The estimates are optimal in the sense that they coincide with classical weighted gradient estimates in the event the coefficients are continuous and the domain has smooth boundary.

Two Linear Noncoercive Dirichlet Problems in Duality

We give a self-contained and simple approach to prove the existence and uniqueness of a weak solution to a linear elliptic boundary value problem with drift in divergence form. Taking advantage of the method of continuity, we also deal with the relative dual problem.

A two-grid parallel partition of unity finite element scheme

A two-grid parallel partition of unity finite element scheme is proposed and analyzed in this paper for linear elliptic boundary value problems. The interesting features of this scheme contain the following: (1) a partition of unity is constructed to derive the globally continuous finite element solution; (2) errors decay exponentially with patches of diameter kH increase; and (3) a global coarse grid correction is done to improve the L2 − accuracy of the approximation. Numerical experiments are presented at the end of the paper to support our analysis.

Elliptic boundary value problems with Gaussian white noise loads

Linear second order elliptic boundary value problems (BVP) on bounded Lipschitz domains are studied in the case of Gaussian white noise loads. The challenging cases of Neumann and Robin BVPs are considered.The main obstacle for usual variational methods is the irregularity of the load. In particular, the Neumann boundary values are not well-defined.In this work, the BVP is formulated by replacing the continuity of boundary trace mappings with measurability. Instead of variational methods alone, the novel BVP derives also from Cameron–Martin space techniques.The new BVP returns the study of irregular white noise to the study of L2-loads.

Meshless analysis of an improved element-free Galerkin method for linear and nonlinear elliptic problems**Project supported by the National Natural Science Foundation of China (Grant No. 11471063), the Chongqing Research Program of Basic Research and Frontier Technology, China (Grant No. cstc2015jcyjBX0083), and the Educational Commission Foundation of Chongqing City, China (Grant No. KJ1600330).

We first give a stabilized improved moving least squares (IMLS) approximation, which has better computational stability and precision than the IMLS approximation. Then, analysis of the improved element-free Galerkin method is provided theoretically for both linear and nonlinear elliptic boundary value problems. Finally, numerical examples are given to verify the theoretical analysis.

POD-Galerkin reduced-order modeling with adaptive finite element snapshots

We consider model order reduction by proper orthogonal decomposition (POD) for parametrized partial differential equations, where the underlying snapshots are computed with adaptive finite elements. We address computational and theoretical issues arising from the fact that the snapshots are members of different finite element spaces. We propose a method to create a POD-Galerkin model without interpolating the snapshots onto their common finite element mesh. The error of the reduced-order solution is not necessarily Galerkin orthogonal to the reduced space created from space-adapted snapshot. We analyze how this influences the error assessment for POD-Galerkin models of linear elliptic boundary value problems. As a numerical example we consider a two-dimensional convection–diffusion equation with a parametrized convective direction. To illustrate the applicability of our techniques to non-linear time-dependent problems, we present a test case of a two-dimensional viscous Burgers equation with parametrized initial data.

Elliptic boundary value problems in spaces of continuous functions

In these notes we consider second order linear elliptic boundary value problems in the framework of different spaces on continuous functions. We appeal to a general formulation which contains some interesting particular cases as, for instance, a new class of functional spaces, called here Holog spaces and denoted by the symbol $\, C^{0,\,\lambda}_\alpha(\overline{\Omega}) \,,$ $\,0 \leq\,\lambda<\,1\,,$ and $\,\alpha \in\,\mathbb{R}\,.$ One has the following inclusions $$ C^{0,\,\lambda+\,\epsilon}(\overline{\Omega})\subset \,C^{0,\,\lambda}_\alpha(\overline{\Omega})\subset \,C^{0,\,\lambda}(\overline{\Omega}) \subset \,C^{0,\,\lambda,}_{-\alpha}(\overline{\Omega}) \subset\,C^{0,\,\lambda-\,\epsilon}(\overline{\Omega})\,, $$ for $\,\alpha>\,0\,$ ($\epsilon >\,0\,$ arbitrarily small). Roughly speaking, for each fixed $\,\lambda\,,$ the family $\, C^{0,\,\lambda}_\alpha(\overline{\Omega}) \,$ is a refinement of the single Holder classical space $\, C^{0,\,\lambda}(\overline{\Omega})=\,C^{0,\,\lambda}_0(\overline{\Omega})\,.$ On the other hand, for $\,\lambda=\,0\,$ and $\,\alpha>\,0\,,$ $\,C^{0,\,0}_\alpha(\overline{\Omega})=\,\, D^{0,\,\alpha}(\overline{\Omega})\,$ is a Log space. The more interesting feature is that, as for classical Holder (and Sobolev) spaces, full regularity occurs. namely, for each $\,\lambda>\,0\,$ and arbitrary real $\,\alpha\,,$ $\,\nabla^2\,u $ and $\,f\,$ enjoy the same $\, C^{0,\,\lambda}_\alpha(\overline{\Omega}) \,$ regularity. All the above setup is presented as part of a more general picture.

The scaled boundary finite element method for the analysis of 3D crack interaction

The scaled boundary finite element method (SBFEM) can be applied to solve linear elliptic boundary value problems when a so-called scaling center can be defined such that every point on the boundary is visible from this scaling center. From a more practical point of view, this means that in linear elasticity, a separation of variables ansatz can be used for the displacements in a scaled boundary coordinate system. This approach allows an analytical treatment of the problem in the scaling direction. Only the boundary needs to be discretized with finite elements. Employment of the separation of variables ansatz in the virtual work balance yields a Cauchy–Euler differential equation system of second order which can be transformed into an eigenvalue problem and solved by standard eigenvalue solvers for nonsymmetric matrices. A further obtained linear equation system serves for enforcing the boundary conditions. If the scaling center is located directly at a singular point, elliptic boundary value problems containing singularities can be solved with high accuracy and computational efficiency. The application of the SBFEM to the linear elasticity problem of two meeting inter-fiber cracks in a composite laminate exposed to a simple homogeneous temperature decrease reveals the presence of hypersingular stresses.

Application of the method of fundamental solutions to 2D and 3D Signorini problems

This paper presents an application of the method of fundamental solutions (MFS) for the numerical solution of 2D and 3D Signorini problems. In our application, by using a projection technique to tackle the nonlinear Signorini boundary inequality conditions, the original Signorini problem is transformed into a sequence of linear elliptic boundary value problems and then solved by the MFS. Convergence and efficiency of the present MFS is proved theoretically and verified numerically.

Programming finite element methods with weighted B-splines

We discuss the main components of the recently developed FEMB program package, which implements finite element methods with weighted B-splines for basic linear elliptic boundary value problems in two and three dimensions. A three-dimensional implementation without topological restrictions has not been available before. We describe in particular the mathematical background for the recursive quadrature/cubature over boundary cells and explain how to utilize the regular data structure of uniform B-splines efficiently. Considering the Lamé–Navier equations of linear elasticity as a typical example, we illustrate the performance of the main FEMB routines. The numerical tests confirm that, due to the new integration routines, the weighted B-spline solvers have become considerably more efficient.