Discretization Of Elliptic Problems Research Articles

Variational Approaches to a Discrete Elliptic Problem with a Weight

In this article, we are concerned with the existence of at least one, two and three distinct solutions for discrete boundary value problems driven by the Laplacian. The proof of the main result depends on variational methods. By using a consequence of the local minimum theorem due Bonanno we investigate the existence of at least one solution and two solutions for the problem with the weight. Furthermore, by using two critical point theorems, one due Averna and Bonanno, and another due Bonanno we explore the existence of two and three solutions for the problem. We also provide two examples in order to illustrate the main results.

Multiple Existence Results of Nontrivial Solutions for a Class of Second-Order Partial Difference Equations

In this paper, we consider the existence and multiplicity of nontrivial solutions for discrete elliptic Dirichlet problems Δ12u(i−1,j)+Δ22u(i,j−1)=−f((i,j),u(i,j)),(i,j)∈Ω,u(i,0)=u(i,T2+1)=0i∈Z(1,T1),u(0,j)=u(T1+1,j)=0j∈Z(1,T2), which have a symmetric structure. When the nonlinearity f(·,u) is resonant at both zero and infinity, we construct a variational functional on a suitable function space and turn the problem of finding nontrivial solutions of discrete elliptic Dirichlet problems to seeking nontrivial critical points of the corresponding functional. We establish a series of results based on the existence of one, two or five nontrivial solutions under reasonable assumptions. Our results depend on the Morse theory and local linking.

Existence and multiplicity of nontrivial solutions to discrete elliptic Dirichlet problems

<abstract><p>In this paper, we study discrete elliptic Dirichlet problems. Applying a variational technique together with Morse theory, we establish several results on the existence and multiplicity of nontrivial solutions. Finally, two examples and numerical simulations are provided to illustrate our theoretical results.</p></abstract>

Uniform Subspace Correction Preconditioners for Discontinuous Galerkin Methods with hp-Refinement

In this paper, we develop subspace correction preconditioners for discontinuous Galerkin (DG) discretizations of elliptic problems with $hp$-refinement. These preconditioners are based on the decomposition of the DG finite element space into a conforming subspace, and a set of small nonconforming edge spaces. The conforming subspace is preconditioned using a matrix-free low-order refined technique, which in this work we extend to the $hp$-refinement context using a variational restriction approach. The condition number of the resulting linear system is independent of the granularity of the mesh $h$, and the degree of polynomial approximation $p$. The method is amenable to use with meshes of any degree of irregularity and arbitrary distribution of polynomial degrees. Numerical examples are shown on several test cases involving adaptively and randomly refined meshes, using both the symmetric interior penalty method and the second method of Bassi and Rebay (BR2).

A note on the fast direct method for discrete elliptic problems

Abstract We consider a method due to P. Vassilevski and Yu. A. Kuznetsov [4, 10] for solving linear systems with matrices of low Kronecker rank such that all factors in Kronecker products are banded. Most important examples of such matrices arise from discretized div K grad operator with diffusion term k1(x)k2(y)k3(z). Several practical issues are addressed: an MPI implementation with distribution of data along processor grid inheriting Cartesian 3D structure of discretized problem; implicit deflation of the known nullspace of the system matrix; links with two-grid framework of multigrid algorithm which allow one to remove the requirement of Kronecker structure in one or two of axes. Numerical experiments show the efficiency of 3D data distribution having the scalability analogous to (structured) HYPRE solvers yet the absolute timings being an order of magnitude lower, on the range from 10 to 104 cores.

Computational recovery of time-dependent volatility from integral observations in option pricing

In this paper robust algorithms for numerical identification of time dependent volatility by integral observations of one- and two-asset Black–Scholes models are developed. An average linearization in time of diffusion terms of the discrete initial boundary value problems is used. Then, a decomposition with respect to the volatility of the approximate solution is applied so that the transition to a new time layer is carried out by solving standard discrete elliptic problems. Numerical experiments using simulated as well as real data confirm the effectiveness of the present approach.

A mixed discretization of elliptic problems on polyhedra using anisotropic hybrid meshes

A virtual element method is introduced for the mixed approximation of a simple model problem for the Laplace operator on a polyhedron. The method is fully analysed when the meshes are made up of triangular right prisms, pyramids and tetrahedra. The local discrete spaces coincide with the lowest order Raviart–Thomas spaces on tetrahedral and triangular right prismatic elements, and extend them to pyramidal elements. The discrete scheme is well posed and optimal interpolation error estimates are proved on meshes which allow for anisotropic elements. In particular, local interpolation error estimates for the discrete element space are optimal and anisotropic on anisotropic right prisms. Furthermore, a discretization of the model problem in the presence of edge and vertex singularities is analysed for the proposed method on a family of suitably designed graded meshes, and optimal estimates for the approximation error are obtained, extending in this way the results of Farhloul et al. (ESAIM Math Model Numer Anal 35:907–920, 2001) where cylindrical domains with edge singularities were considered.

Incomplete Iterative Implicit Schemes

Abstract In numerical solving boundary value problems for parabolic equations, two- or three-level implicit schemes are in common use. Their computational implementation is based on solving a discrete elliptic problem at a new time level. For this purpose, various iterative methods are applied to multidimensional problems evaluating an approximate solution with some error. It is necessary to ensure that these errors do not violate the stability of the approximate solution, i.e., the approximate solution must converge to the exact one. In the present paper, these questions are investigated in numerical solving a Cauchy problem for a linear evolutionary equation of first order, which is considered in a finite-dimensional Hilbert space. The study is based on the general theory of stability (well-posedness) of operator-difference schemes developed by Samarskii. The iterative methods used in the study are considered from the same general positions.

Efficient Operator-Coarsening Multigrid Schemes for Local Discontinuous Galerkin Methods

An efficient $hp$-multigrid scheme is presented for local discontinuous Galerkin (LDG) discretizations of elliptic problems, formulated around the idea of separately coarsening the underlying discr...

A cryptosystem based on chaotic maps and factoring problems

A cryptosystem allows a sender to send any confidential or private message using a receiver's public key and the receiver next confirms the integrity and validity of the received message using his own secret key. Cryptosystem algorithms can be categorised based on the type of security suppositions, for example discrete logarithm, factorisation, and elliptic curve hard problems, which are all currently believed to be unsolvable in a reasonable time of period. Recently, cryptosystems based on chaotic maps have been proposed. Due to some subtle and close relationship between the properties of traditional cryptosystems and chaotic-based systems, the idea of using chaotic in cryptography has received a great deal of attention from many cryptography's researchers. Therefore, to enhance system security, we explore the implementation of a cryptosystem algorithm based on both cryptographic and chaotic system characteristics. We also provide security against known cryptographic attacks and discuss the performance analysis of the developed system.

A Two-Level Overlapping Hybrid Domain Decomposition Method for Eigenvalue Problems

In this paper, we present a two-level overlapping hybrid domain decomposition method for solving the large scale discrete elliptic eigenvalue problems. In order to eliminate the components in the o...

On convergence of the WR-HSS iteration method for a system of linear differential equations

We consider a waveform relaxation (WR) method based on the Hermitian/skew-Hermitian splitting (HSS) of the system matrices, which is a continuous-time iteration method. In actual implementation, the continuous-time WR-HSS method (CWR-HSS) is replaced by the discrete-time WR-HSS method (DWR-HSS) defined on a time-level-sequence. If the time-step-size tends to zero, the approximate solution obtained by the DWR-HSS method on each time level is proved to converge to the limit of the approximate solution obtained by the CWR-HSS method, i.e., the exact solution of a corresponding system of linear differential equations. Finally, the above relationship between the CWR-HSS method and the DWR-HSS method is verified by the numerical tests based on the unsteady discrete elliptic problem. Therefore, the DWR-HSS method is a reliable option for solving the unsteady discrete elliptic problem in both theoretical and practical aspects.

A secure ID-based signcryption scheme based on elliptic curve cryptography

Signcryption is a cryptographic primitive which at the same time give both the capacity of digital signature and public key encryption in a single logical step. Identity based cryptography is a distinct option for the traditional certificate based cryptosystem. Its principal thought is that every client utilises his identity information as his public key. Elliptic curve cryptosystem (ECC) have new received consistent attention because of their higher security per bit as compare to other cryptosystem. This paper presents a new identity based signcryption based on elliptic curve cryptography. Its security is dependent on elliptic curve discrete logarithm problem (ECDLP) and elliptic curve Diffie-Hellman problem (ECDHP). The proposed scheme can be very useful in low-end resource devices such as mobile communication, mobile banking, personal digital assistant (PDA) and internet of things (IoT).

A fast immersed boundary method for external incompressible viscous flows using lattice Green's functions

A new parallel, computationally efficient immersed boundary method for solving three-dimensional, viscous, incompressible flows on unbounded domains is presented. Immersed surfaces with prescribed motions are generated using the interpolation and regularization operators obtained from the discrete delta function approach of the original (Peskin's) immersed boundary method. Unlike Peskin's method, boundary forces are regarded as Lagrange multipliers that are used to satisfy the no-slip condition. The incompressible Navier–Stokes equations are discretized on an unbounded staggered Cartesian grid and are solved in a finite number of operations using lattice Green's function techniques. These techniques are used to automatically enforce the natural free-space boundary conditions and to implement a novel block-wise adaptive grid that significantly reduces the run-time cost of solutions by limiting operations to grid cells in the immediate vicinity and near-wake region of the immersed surface. These techniques also enable the construction of practical discrete viscous integrating factors that are used in combination with specialized half-explicit Runge–Kutta schemes to accurately and efficiently solve the differential algebraic equations describing the discrete momentum equation, incompressibility constraint, and no-slip constraint. Linear systems of equations resulting from the time integration scheme are efficiently solved using an approximation-free nested projection technique. The algebraic properties of the discrete operators are used to reduce projection steps to simple discrete elliptic problems, e.g. discrete Poisson problems, that are compatible with recent parallel fast multipole methods for difference equations. Numerical experiments on low-aspect-ratio flat plates and spheres at Reynolds numbers up to 3700 are used to verify the accuracy and physical fidelity of the formulation.

The WR–HSS iteration method for a system of linear differential equations and its applications to the unsteady discrete elliptic problem

We consider the numerical method for non-self-adjoint positive definite linear differential equations, and its application to the unsteady discrete elliptic problem, which is derived from spatial discretization of the unsteady elliptic problem with Dirichlet boundary condition. Based on the idea of the alternating direction implicit (ADI) iteration technique and the Hermitian/skew-Hermitian splitting (HSS), we establish a waveform relaxation (WR) iteration method for solving the non-self-adjoint positive definite linear differential equations, called the WR–HSS method. We analyze the convergence property of the WR–HSS method, and prove that the WR–HSS method is unconditionally convergent to the solution of the system of linear differential equations. In addition, we derive the upper bound of the contraction factor of the WR–HSS method in each iteration which is only dependent on the Hermitian part of the corresponding non-self-adjoint positive definite linear differential operator. Finally, the applications of the WR–HSS method to the unsteady discrete elliptic problem demonstrate its effectiveness and the correctness of the theoretical results.

Wavelet basis of cubic splines on the hypercube satisfying homogeneous boundary conditions

In this paper, we propose a construction of a new cubic spline-wavelet basis on the hypercube satisfying homogeneous Dirichlet boundary conditions. Wavelets have two vanishing moments. Stiffness matrices arising from discretization of elliptic problems using a constructed wavelet basis have uniformly bounded condition numbers and we show that these condition numbers are small. We present quantitative properties of the constructed basis and we provide a numerical example to show the efficiency of the Galerkin method using the constructed basis.

On domain decomposition preconditioner of BPS type for finite element discretizations of 3D elliptic equations

BPS is a well known an efficient and rather general domain decomposition Dirichlet-Dirichlet type preconditioner, suggested in the famous series of papers Bramble, Pasciak and Schatz (1986–1989). Since then, it has been serving as the origin for the whole family of domain decomposition Dirichlet-Dirichlet type preconditioners-solvers as for h so hp discretizations of elliptic problems. For its original version, designed for h discretizations, the named authors proved the bound O(1 + log2H/h) for the relative condition number under some restricting conditions on the domain decomposition and finite element discretization. Here H/h is the maximal relation of the characteristic size H of a decomposition subdomain to the mesh parameter h of its discretization. It was assumed that subdomains are images of the reference unite cube by trilinear mappings. Later similar bounds related to h discretizations were proved for more general domain decompositions, defined by means of coarse tetrahedral meshes. These results, accompanied by the development of some special tools of analysis aimed at such type of decompositions, were summarized in the book of Toselli and Widlund (2005). This paper is also confined to h discretizations. We further expand the range of admissible domain decompositions for constructing BPS preconditioners, in which decomposition subdomains can be convex polyhedrons, satisfying some conditions of shape regularity. We prove the bound for the relative condition number with the same dependence on H/h as in the bound given above. Along the way to this result, we simplify the proof of the so called abstract bound for the relative condition number of the domain decomposition preconditioner. In the part, related to the analysis of the interface sub-problem preconditioning, our technical tools are generalization of those used by Bramble, Pasciak and Schatz.

A response to a critique of “A novel electronic cash system with trustee-based anonymity revocation from pairing,” by Chen, Chou, Sun and Cho (2011)

We provide a brief rebuttal of Chang (2012), who suggested that there were flaws in our article, Chen et al. (2011), that deserved further investigation and comment. We believe that these criticisms are unfounded, and offer some additional details related to the intractable discrete elliptic curve discrete logarithm problem to further support the case we wish to make.

Parallel numerical solution of the time‐harmonic Maxwell equations in mixed form

SUMMARYWe develop a fully scalable parallel implementation of an iterative solver for the time‐harmonic Maxwell equations with vanishing wave numbers. We use a mixed finite element discretization on tetrahedral meshes, based on the lowest order Nédélec finite element pair of the first kind. We apply the block diagonal preconditioning approach of Greif and Schötzau (Numer. Linear Algebra Appl. 2007; 14(4):281–297), and use the nodal auxiliary space preconditioning technique of Hiptmair and Xu (SIAM J. Numer. Anal. 2007; 45(6):2483–2509) as the inner iteration for the shifted curl–curl operator. Algebraic multigrid is employed to solve the resulting sequence of discrete elliptic problems. We demonstrate the performance of our parallel solver on problems with constant and variable coefficients. Our numerical results indicate good scalability with the mesh size on uniform, unstructured, and locally refined meshes. Copyright © 2011 John Wiley & Sons, Ltd.

Uniformly Convergent Iterative Methods for Discontinuous Galerkin Discretizations

We present iterative and preconditioning techniques for the solution of the linear systems resulting from several discontinuous Galerkin (DG) Interior Penalty (IP) discretizations of elliptic problems. We analyze the convergence properties of these algorithms for both symmetric and non-symmetric IP schemes. The iterative methods are based on a "natural" decomposition of the first order DG finite element space as a direct sum of the Crouzeix-Raviart non-conforming finite element space and a subspace that contains functions discontinuous at interior faces. We also present numerical examples confirming the theoretical results.