Sobolev Equations Research Articles

Meshless RBFs method for numerical solutions of two-dimensional high order fractional Sobolev equations

In this paper, meshless RBFs method is proposed to solve two-dimensional time-fractional Sobolev equations. The proposed method uses RBFs for approximation of spatial operator. Finite difference formula of O(δt2−α)(0<α≤1) is used for time-fractional derivative approximation while θ-rule (0≤θ≤1) as time stepping scheme for the advancement of solution in time. Validation of the proposed method is made by considering various test examples from literature. Simulated results are found in very good agreement with available exact solutions. A rigorous comparative analysis made with other methods testifies proposed method’s superiority in higher dimensions. Efficiency and accuracy of RBFs method are examined by varying number of nodes N in the domain of influence, time-step size δt, as well as L2, L∞ and Lrms error norms. Linearized stability analysis of the proposed method is thoroughly discussed and verified numerically to support the analysis.

On adaptive mesh for the initial boundary value singularly perturbed delay Sobolev problems

AbstractA uniform finite difference method on a B‐mesh is applied to solve the initial‐boundary value problem for singularly perturbed delay Sobolev equations. To solve the foresold problem, finite difference scheme on a special nonuniform mesh, whose solution converges point‐wise independently of the singular perturbation parameter is constructed and analyzed. The present paper also aims at discussing the stability and convergence analysis of the method. An error analysis shows that the method is of second order convergent in the discrete maximum norm independent of the perturbation parameter. A numerical example and the simulation results show the effectiveness of our theoretical results.

A block-centered finite difference method for the nonlinear Sobolev equation on nonuniform rectangular grids

In this article, a block-centered finite difference method for the nonlinear Sobolev equation is introduced and analyzed. The stability and the global convergence of the scheme are proved rigorously. Some a priori estimates of discrete norms with superconvergence O(Δt+h2+k2) for scalar unknown p, its gradient u and its flux q are established on nonuniform rectangular grids, where Δt, h and k are the step sizes in time, space in x- and y-direction. Moreover, the applicability and accuracy of the scheme are demonstrated by numerical experiments to support our theoretical analysis.

Application of meshfree spectral method for the solution of multi-dimensional time-fractional Sobolev equations

The present study aims to formulate a meshfree spectral method to numerically solve multi-dimensional time-fractional Sobolev equations. In this method radial basis functions and point interpolation approach is utilized to build meshfree shape functions. These shape functions, having Kronecker delta function property, are then used for spatial discretization. Forward difference formula, in alignment to a quadrature rule, is used for temporal disctrezation. Validation of the proposed method is made by considering various test examples from literature. Simulated results reveal very good agreement with available exact solutions which are reported in graphical and tabulated forms. Approximation quality and efficiency of the current method is achieved in terms of e2, e∞ and erms error norms, number of collocation points as well as time-step size. Stability of the proposed method is thoroughly analyzed and discussed.

A reduced-order extrapolating collocation spectral method based on POD for the 2D Sobolev equations

In this paper, we mainly use a proper orthogonal decomposition (POD) to reduce the order of the coefficient vectors of the solutions for the classical collocation spectral (CS) method of two-dimensional (2D) Sobolev equations. We first establish a reduced-order extrapolating collocation spectral (ROECS) method for 2D Sobolev equations so that the ROECS method includes the same basis functions as the classic CS method and the superiority of the classic CS method. Then we use the matrix means to discuss the existence, stability, and convergence of the ROECS solutions so that the procedure of theoretical analysis becomes very concise. Lastly, we present two set of numerical examples to validate the effectiveness of theoretical conclusions and to illuminate that the ROECS method is far superior to the classic CS method, which shows that the ROECS method is quite valid to solve Sobolev equations. Therefore, both theory and method of this paper are completely different from the existing reduced-order methods.

Superconvergence analysis of a linearized three‐step backward differential formula finite element method for nonlinear Sobolev equation

A linearized three‐step backward differential formula (BDF) Galerkin finite element method (FEM) is developed for nonlinear Sobolev equation with bilinear element. Temporal error and spatial error are discussed through introducing a time‐discrete system. Solutions of the time‐discrete system are bounded in H2‐norm by the temporal error. Superconvergence results of order O(h2 + τ3) in H1‐norm for the original variable are deduced based on the spatial error. Some new tricks are utilized to get higher order of the temporal error and the spatial error. At last, two numerical examples are provided to support the theoretical analysis. Here, h is the subdivision parameter, and τ is the time step.

A mass-conservative characteristic splitting mixed finite element method for convection-dominated Sobolev equation

In this article, a new characteristic splitting mixed finite element method is proposed for solving convection-dominated Sobolev equation. In this algorithm the mass-conservative characteristic (MCC) scheme is applied to approximate the time derivative term plus the convection term, and the splitting mixed finite element (SMFE) technique is used to approximate the primal unknown function and the introduced unknown flux. This procedure not only keeps the mass balance but also results in a splitting symmetric positive definite mixed system where we do not need to solve a coupled system. The convergence analysis is considered and the corresponding optimal error estimates in L2 norm for the primal unknown and in H(div) norm for the unknown flux are derived, respectively. Numerical examples are provided to confirm the theoretical results.

On Some Discrete Potential Like Operators

Abstract We consider some discrete pseudo-differential equations in discrete Sobolev–Slobodetskii spaces. For a discrete half-space and certain values of an index of periodic factorization for an elliptic symbol we introduce additional potential-like unknowns and prove existence and uniqueness theorem in appropriate discrete Sobolev–Slobodetskii spaces.

Crank–Nicolson Finite Element Scheme and Modified Reduced-Order Scheme for Fractional Sobolev Equation

In this paper, a Crank–Nicolson finite difference/finite element method is considered to obtain the numerical solution for a time fractional Sobolev equation. Firstly, the classical finite element method is presented. Stability and error estimation for the fully discrete scheme are rigorously established. However, the amount of calculation and computing time are too large due to many degrees of freedom of classical finite element scheme and nonlocality of fractional differential operator. And then the modified reduced-order finite element scheme with low dimensions and sufficiently high accuracy, which is based on proper orthogonal decomposition technique, is provided. Stability and convergence for the reduced-order scheme are also studied. At last, numerical examples show that the results of numerical computation are consistent with previous theoretical conclusions.

An expanded mixed finite element method for two-dimensional Sobolev equations

In this paper we provide an expanded mixed finite element method for a class of two-dimensional Sobolev equations. The optimal error estimates for a semi-discrete scheme and a fully discrete scheme are obtained. Also numerical examples are stated to verify our theoretical results.

Fully discrete approximation of general nonlinear Sobolev equations

We consider abstract quasilinear evolution equations of Sobolev type in a Hilbert setting. We propose two fully discrete schemes and prove some error estimates under minimal assumptions. Various examples that enter into our abstract framework are considered, for each of them our theoretical results are confirmed by several numerical experiments.

Asymptotic Behavior at Infinity of Solutions to the Nonhomogeneous Sobolev Equation

Under study is the asymptotic behavior at infinity of solutions to the Cauchy problem to the nonhomogeneous Sobolev equation. We obtain the form of the limit function and the convergence rate.

Local Discontinuous Galerkin Methods with Generalized Alternating Numerical Fluxes for Two-dimensional Linear Sobolev Equation

In this paper we present an efficient and high-order numerical method to solve two-dimensional linear Sobolev equations, which is based on the local discontinuous Galerkin (LDG) method with the upwind-biased fluxes and generalized alternating fluxes. A weak stability is given for both schemes, and a strong stability is established if the initial solutions exactly satisfy the elemental discontinuous Galerkin discretization. Moreover, the sharp error estimate in $$L^2$$ -norm is established, by an elaborate application of the generalized Gauss–Radau projection. A fully-discrete LDG scheme is also considered, where the third-order explicit TVD Runge–Kutta algorithm is adopted. Finally some numerical experiments are given.

Two-grid finite element methods combined with Crank-Nicolson scheme for nonlinear Sobolev equations

In this paper, we present a second-order accurate Crank-Nicolson scheme for the two-grid finite element methods of the nonlinear Sobolev equations. This method involves solving a small nonlinear system on a coarse mesh with mesh size H and a linear system on a fine mesh with mesh size h, which can still maintain the asymptotically optimal accuracy compared with the standard finite element method. However, the two-grid scheme can reduce workload and save a lot of CPU time. The optimal error estimates in H1-norm show that the two-grid methods can achieve optimal convergence order when the mesh sizes satisfy h = O(H2). These estimates are shown to be uniform in time. Numerical results are provided to verify the theoretical estimates.

Blow-up solutions for Hardy–Sobolev equations on compact Riemannian manifolds

Let (M, g) be a smooth compact Riemannian manifold of dimension $$n\ge 6$$ , $$\xi _0\in M$$ , and we are concerned with the following Hardy–Sobolev elliptic equations: 0.1 $$\begin{aligned} -\Delta _gu+h(x)u=\frac{u^{2^{*}(s)-1-\epsilon }}{d_{g}(x,\xi _0)^s},\ \ \ \ u>0\ \ \mathrm{in} \ \ M, \end{aligned}$$ where $$\Delta _g\,=\,\mathrm{div}_g(\nabla )$$ is the Laplace–Beltrami operator on M, h(x) is a $$C^1$$ function on M, $$\epsilon $$ is a sufficiently small real parameter, $$2^{*}(s):=\frac{2(n-s)}{n-2}$$ is the critical Hardy–Sobolev exponent with $$s\in (0,2)$$ , and $$d_{g}$$ is the Riemannian distance on M. Performing the Lyapunov–Schmidt reduction procedure, we obtain the existence of blow-up families of positive solutions of problem (0.1).

A collocation spectral method for two-dimensional Sobolev equations

This article mainly studies a collocation spectral method for two-dimensional (2D) Sobolev equations. To this end, a collocation spectral model based on the Chebyshev polynomials for the 2D Sobolev equations is first established. And then, the existence, uniqueness, stability, and convergence of the collocation spectral numerical solutions are discussed. Finally, some numerical experiments are provided to verify the corrections of theoretical results. This implies that the collocation spectral model is very effective for solving the 2D Sobolev equations.

A reduced-order extrapolated finite difference iterative scheme based on POD method for 2D Sobolev equation

In this study, we devote ourselves to the reduced-order extrapolated finite difference iterative (ROEFDI) modeling and analysis for the two-dimensional (2D) Sobolev equation. To this end, we first establish the reduced-order extrapolated finite difference iterative (ROEFDI) scheme holding sufficiently high accuracy but containing very few degrees of freedom for the 2D Sobolev equation via the proper orthogonal decomposition (POD) technique. And then, we analyze the stability and convergence of the ROEFDI solutions. Finally, we use the numerical experiments to verify the feasibility and effectiveness of the ROEFDI scheme.

A Newton linearized compact finite difference scheme for one class of Sobolev equations

In this article, a Newton linearized compact finite difference scheme is proposed to numerically solve a class of Sobolev equations. The unique solvability, convergence, and stability of the proposed scheme are proved. It is shown that the proposed method is of order 2 in temporal direction and order 4 in spatial direction. Moreover, compare to the classical extrapolated Crank‐Nicolson method or the second‐order multistep implicit–explicit methods, the proposed scheme is easier to be implemented as it only requires one starting value. Finally, numerical experiments on one and two‐dimensional problems are presented to illustrate our theoretical results.

Asymptotic Analysis and Optimal Error estimates for Benjamin‐Bona‐Mahony‐Burgers' Type Equations

In this article, stabilization result for the Benjamin‐Bona‐Mahony‐Burgers' (BBM‐B) equation, that is, convergence of unsteady solution to steady state solution is established under the assumption that a linearized steady state eigenvalue problem has a minimal positive eigenvalue. Based on appropriate conditions on the forcing function, exponential decay estimates in , and ‐norms are derived, which are valid uniformly with respect to the coefficient of dispersion as it tends to zero. It is, further, observed that the decay rate for the BBM‐B equation is smaller than that of the decay rate for the Burgers equation. Then, a semidiscrete Galerkin method for spatial direction keeping time variable continuous is considered and stabilization results are discussed for the semidiscrete problem. Moreover, optimal error estimates in ‐norms preserving exponential decay property are established using the steady state error estimates. For a complete discrete scheme, a backward Euler method is applied for the time discretization and stabilization results are again proved for the fully discrete problem. Subsequently, numerical experiments are conducted, which verify our theoretical results. The article is finally concluded with a brief discussion on an extension to a multidimensional nonlinear Sobolev equation with Burgers' type nonlinearity.

Hardy–Sobolev equations with asymptotically vanishing singularity: Blow-up analysis for the minimal energy

We study the asymptotic behavior of a sequence of positive solutions (uϵ)ϵ>0 as ϵ→0 to the family of equations Δuϵ+a(x)uϵ=uϵ2∗(sϵ)−1|x|sϵ in Ωuϵ=0 on ∂Ω.where (sϵ)ϵ>0 is a sequence of positive real numbers such that limϵ→0sϵ=0, 2∗(sϵ)≔2(n−sϵ)n−2 and Ω⊂Rn is a bounded smooth domain such that 0∈∂Ω. When the sequence (uϵ)ϵ>0 is uniformly bounded in L∞, then up to a subsequence it converges strongly to a minimizing solution of the stationary Schrödinger equation with critical growth. In case the sequence blows up, we obtain strong pointwise control on the blow-up sequence, and then using the Pohozaev identity localize the point of singularity, which in this case can at most be one, and derive precise blow-up rates. In particular when n=3 or a≡0 then blow-up can occur only at an interior point of Ω or the point 0∈∂Ω.