Elliptic Boundary Value Problems Research Articles

The Scaling Limit of the $$(\nabla +\Delta )$$-Model

In this article we study the scaling limit of the interface model on \({{\,\mathrm{{\mathbb {Z}}}\,}}^d\) where the Hamiltonian is given by a mixed gradient and Laplacian interaction. We show that in any dimension the scaling limit is given by the Gaussian free field. We discuss the appropriate spaces in which the convergence takes place. While in infinite volume the proof is based on Fourier analytic methods, in finite volume we rely on some discrete PDE techniques involving finite-difference approximation of elliptic boundary value problems.

Symmetry breaking bifurcations for two overdetermined boundary value problems with non-constant Neumann condition on exterior domains in

We study two overdetermined elliptic boundary value problems on exterior domains (the complement of a ball and the complement of a solid cylinder in respectively). The Neumann condition is non-constant and involves the mean curvature of the boundary. We show there exists a family of bifurcation branches of domains which are small deformations of the complement of a ball and of the complement of a solid cylinder, respectively, and which support the solution of the overdetermined boundary value problem.

Boundary value problems of elliptic and parabolic type with boundary data of negative regularity

We study elliptic and parabolic boundary value problems in spaces of mixed scales with mixed smoothness on the half-space. The aim is to solve boundary value problems with boundary data of negative regularity and to describe the singularities of solutions at the boundary. To this end, we derive mapping properties of Poisson operators in mixed scales with mixed smoothness. We also derive mathcal {R}-sectoriality results for homogeneous boundary data in the case that the smoothness in normal direction is not too large.

Efficient Numerical Solution of the EMI Model Representing the Extracellular Space (E), Cell Membrane (M) and Intracellular Space (I) of a Collection of Cardiac Cells

The EMI model represents excitable cells in a more accurate manner than traditional homogenized models at the price of increased computational complexity. The increased complexity of solving the EMI model stems from a significant increase in the number of computational nodes and from the form of the linear systems that need to be solved. Here, we will show that the latter problem can be solved by careful use of operator splitting of the spatially coupled equations. By using this method, the linear systems can be broken into sub-problems that are of the classical type of linear, elliptic boundary value problems. Therefore, the vast collection of methods for solving linear, elliptic partial differential equations can be used. We demonstrate that this enables us to solve the systems using shared-memory parallel computers. The computing time scales perfectly with the number of physical cells. For a collection of 512×256 cells, we solved linear systems with about2.5×108unknows. Since the computational effort scales linearly with the number of physical cells, we believe that larger computers can be used to simulate millions of excitable cells and thus allow careful analysis of physiological systems of great importance.

Interface Immersed Particle Difference Method for weak discontinuity in elliptic boundary value problems

In this paper, the Interface Immersed Particle Difference Method (IIPDM) for weak discontinuity in elliptic problems is presented. Heat conduction and potential flow problems with smooth and non-smooth interfaces are considered. The previously developed Particle Difference Method (PDM) accurately solves this class of interfacial singularity problems, however, it requires additional difference equations on the interfacial points. In contrast, the IIPDM no longer requires the additional interfacial equations since the interface condition is already immersed in the particle derivative approximation through the moving least squares procedure. The method successfully captures both singularities and discontinuities in the derivative field due to the interface geometry regardless of its smoothness. In fact, enforcement of the interface condition is conducted both in an implicit manner and in an explicit manner enriching the polynomial basis as well as the local approximation. Under the constrained derivative approximation, discretization of the elliptic PDE in its strong form leads to a difference scheme or a point collocation scheme involving only the particles. Consequently, the strong formulation can avoid the increase in total system size and improves the computational efficiency. Numerical experiments show that the IIPDM can sharply capture discontinuities and singularities within a solution field. Furthermore, convergence studies of various elliptic interface problems demonstrate efficiencies captured from the IIPDM when comparing convergence rates against the PDM.

An Elliptic Boundary Value Problem with Fractional Nonlinearity

We investigate the existence and uniqueness of solutions to second-order elliptic boundary value problems containing a power nonlinearity applied to a fractional Laplacian. We detect the critical power separating the existence from the nonexistence regimes. For the existence results, we make use of a particular class of weighted Sobolev spaces to compensate for boundary singularities which are naturally built in the problem.

Space-Time Finite Element Discretization of Parabolic Optimal Control Problems with Energy Regularization

In this paper, we analyze space-time finite element methods for the numerical solution of distributed parabolic optimal control problems with energy regularization in the Bochner space $L^2(0,T;H^{-1}(\Omega))$. By duality, the related norm can be evaluated by means of the solution of an elliptic quasi-stationary boundary value problem. When eliminating the control, we end up with the reduced optimality system that is nothing but the variational formulation of the coupled forward-backward primal and adjoint equations. Using Babuška's theorem, we prove unique solvability in the continuous case. Furthermore, we establish the discrete inf-sup condition for any conforming space-time finite element discretization yielding quasi-optimal discretization error estimates. Various numerical examples confirm the theoretical findings. We emphasize that the energy regularization results in a more localized control with sharper contours for discontinuous target functions, which is demonstrated by a comparison with an $L^2$-regularization and with a sparse optimal control approach.

ON A DISCRETE ELLIPTIC PROBLEM WITH A WEIGHT

<abstract> Using the variational approach and the critical point theory, we established several criteria for the existence of at least one nontrivial solution for a discrete elliptic boundary value problem with a weight <italic>p</italic> (·, ·) and depending on a real parameter λ. </abstract>

Immersed hybrid difference methods for elliptic boundary value problems by artificial interface conditions

&lt;p style='text-indent:20px;'&gt;We propose an immersed hybrid difference method for elliptic boundary value problems by artificial interface conditions. The artificial interface condition is derived by imposing the given boundary condition weakly with the penalty parameter as in the Nitsche trick and it maintains ellipticity. Then, the derived interface problems can be solved by the hybrid difference approach together with a proper virtual to real transformation. Therefore, the boundary value problems can be solved on a fixed mesh independently of geometric shapes of boundaries. Numerical tests on several types of boundary interfaces are presented to demonstrate efficiency of the suggested method.&lt;/p&gt;

Asymptotic analysis of an elliptic boundary-value problem in a wedge

Asymptotic analysis of an elliptic boundary-value problem in a wedge

On solving elliptic boundary value problems using a meshless method with radial polynomials

This paper presents the meshless method using radial polynomials with the combination of the multiple source collocation scheme for solving elliptic boundary value problems. In the proposed method, the basis function is based on the radial polynomials, which is different from the conventional radial basis functions that approximate the solution using the specific function such as the multiquadric function with the shape parameter for infinitely differentiable. The radial polynomial basis function is a non-singular series function in nature which is infinitely smooth and differentiable in nature without using the shape parameter. With the combination of the multiple source collocation scheme, the center point is regarded as the source point for the interpolation of the radial polynomials. Numerical solutions in multiple dimensions are approximated by applying the radial polynomials with given terms of the radial polynomials. The comparison of the proposed method with the radial basis function collocation method (RBFCM) using the multiquadric and polyharmonic spline functions is conducted. Results demonstrate that the accuracy obtained from the proposed method is better than that of the conventional RBFCM with the same number of collocation points. In addition, highly accurate solutions with the increase of radial polynomial terms may be obtained.

Elliptic problems with rough boundary data in generalized Sobolev spaces

We investigate regular elliptic boundary-value problems in boun\-ded domains and show the Fredholm property for the related operators in an extended scale formed by inner product Sobolev spaces (of arbitrary real orders) and corresponding interpolation Hilbert spaces. In particular, we can deal with boundary data with arbitrary low regularity. In addition, we show interpolation properties for the extended scale, embedding results, and global and local a priori estimates for solutions to the problems under investigation. The results are applied to elliptic problems with homogeneous right-hand side and to elliptic problems with rough boundary data in Nikolskii spaces, which allows us to treat some cases of white noise on the boundary.

A Survey on Numerical Methods for Spectral Space-Fractional Diffusion Problems

AbstractThe survey is devoted to numerical solution of the equation $ {\mathcal A}^\alpha u=f $, 0 &lt; α&lt;1, where $ {\mathcal A} $ is a symmetric positive definite operator corresponding to a second order elliptic boundary value problem in a bounded domain Ω in ℝd. The fractional power $ {\mathcal A}^\alpha $ is a non-local operator and is defined though the spectrum of $ {\mathcal A} $. Due to growing interest and demand in applications of sub-diffusion models to physics and engineering, in the last decade, several numerical approaches have been proposed, studied, and tested. We consider discretizations of the elliptic operator $ {\mathcal A} $ by using an N-dimensional finite element space Vh or finite differences over a uniform mesh with N points. In the case of finite element approximation we get a symmetric and positive definite operator $ {\mathcal A}_h: V_h \to V_h $, which results in an operator equation $ {\mathcal A}_h^{\alpha} u_h = f_h $ for uh ∈ Vh.The numerical solution of this equation is based on the following three equivalent representations of the solution: (1) Dunford-Taylor integral formula (or its equivalent Balakrishnan formula, (2.5), (2) extension of the a second order elliptic problem in Ω × (0, ∞)⊂ ℝd+1 [17,55] (with a local operator) or as a pseudo-parabolic equation in the cylinder (x, t) ∈ Ω × (0, 1), [70, 29], (3) spectral representation (2.6) and the best uniform rational approximation (BURA) of zα on [0, 1], [37,40].Though substantially different in origin and their analysis, these methods can be interpreted as some rational approximation of $ {\mathcal A}_h^{-\alpha} $. In this paper we present the main ideas of these methods and the corresponding algorithms, discuss their accuracy, computational complexity and compare their efficiency and robustness.

Diffusion in arrays of obstacles: beyond homogenization

We revisit the classical problem of diffusion of a scalar (or heat) released in a two-dimensional medium with an embedded periodic array of impermeable obstacles such as perforations. Homogenization theory provides a coarse-grained description of the scalar at large times and predicts that it diffuses with a certain effective diffusivity, so the concentration is approximately Gaussian. We improve on this by developing a large-deviation approximation which also captures the non-Gaussian tails of the concentration through a rate function obtained by solving a family of eigenvalue problems. We focus on cylindrical obstacles and on the dense limit, when the obstacles occupy a large area fraction and non-Gaussianity is most marked. We derive an asymptotic approximation for the rate function in this limit, valid uniformly over a wide range of distances. We use finite-element implementations to solve the eigenvalue problems yielding the rate function for arbitrary obstacle area fractions and an elliptic boundary-value problem arising in the asymptotics calculation. Comparison between numerical results and asymptotic predictions confirms the validity of the latter.

Piecewise Smooth Stationary Euler Flows with Compact Support Via Overdetermined Boundary Problems

We construct new stationary weak solutions of the 3D Euler equation with compact support. The solutions, which are piecewise smooth and discontinuous across a surface, are axisymmetric with swirl. The range of solutions we find is different from, and larger than, the family of smooth stationary solutions recently obtained by Gavrilov and Constantin–La–Vicol; in particular, these solutions are not localizable. A key step in the proof is the construction of solutions to an overdetermined elliptic boundary value problem where one prescribes both Dirichlet and (nonconstant) Neumann data.

Positive solutions of superlinear elliptic problems with discontinuous non-linearities

We consider an elliptic boundary-value problem with a homogeneous Dirichlet boundary condition, a parameter and a discontinuous non-linearity. The positive parameter appears as a multiplicative term in the non-linearity, and the problem has a zero solution for any value of the parameter. The non-linearity has superlinear growth at infinity. We prove the existence of positive solutions by a topological method.

Nonuniqueness for a fully nonlinear, degenerate elliptic boundary-value problem in conformal geometry

One way to generalize the boundary Yamabe problem posed by Escobar is to ask if a given metric on a compact manifold with boundary can be conformally deformed to have vanishing σk-curvature in the interior and constant Hk-curvature on the boundary. When restricting to the closure of the positive k-cone, this is a fully nonlinear degenerate elliptic boundary value problem with fully nonlinear Robin-type boundary condition. We prove nonuniqueness for the boundary-value problem σ4-curvature equals zero and constant H4-curvature by using bifurcation results proven by Case, Moreira and Wang. Surprisingly, our construction via products of sphere and hyperbolic space only works for a finite set of dimensions.

Lyapunov, Hartman-Wintner and De La Vallée Poussin-type inequalities for fractional elliptic boundary value problems

ABSTRACT In this paper, we show Lyapunov and Hartman-Wintner-type inequalities for a fractional partial differential equations with Dirichlet conditions and we give some applications of these inequalities for the eigenvalue problem. Also, we give de La Vallée Poussin-type inequality for the fractional elliptic boundary value problem and Lyapunov-type inequalities for the fractional elliptic systems with Dirichlet conditions.

A remark on elliptic differential equations on manifold

For elliptic boundary value problems of nonlocal type in Euclidean space, the well posedness has been studied by several authors and it has been well understood. On the other hand, such kind of problems on manifolds have not been studied yet. Present article considers differential equations on smooth closed manifolds. It establishes the well posedness of nonlocal boundary value problems of elliptic type, namely Neumann-Bitsadze-Samarskii type nonlocal boundary value problem on manifolds and also DirichletBitsadze-Samarskii type nonlocal boundary value problem on manifolds, in H¨older spaces. In addition, in various H¨older norms, it establishes new coercivity inequalities for solutions of such elliptic nonlocal type boundary value problems on smooth manifolds.

Application of boundary integral equation method to numerical solution of elliptic boundary-value problems in R3

In this paper we propose numerical methods for solving interior and exterior boundary-value problems for the Helmholtz and Laplace equations in complex three-dimensional domains. The method is based on their reduction to boundary integral equations in R2. Using the potentials of the simple and double layers, we obtain boundary integral equations of the Fredholm type with respect to unknown density for Dirichlet and Neumann boundary value problems. As a result of applying integral equations along the boundary of the domain, the dimension of problems is reduced by one. In order to approximate solutions of the obtained weakly singular Fredholm integral equations we suggest general numerical method based on spline approximation of solutions and on the use of adaptive cubatures that take into account the singularities of the kernels. When constructing cubature formulas, essentially non-uniform graded meshes are constructed with grading exponent that depends on the smoothness of the input data. The effectiveness of the method is illustrated with some numerical experiments.