Semidiscrete Solution Research Articles

Asymptotic behavior and finite element error estimates of Kelvin-Voigt viscoelastic fluid flow model

In this article, the convergence of the solution of the Kelvin-Voigt viscoelastic fluid flow model to its steady state solution with exponential rate is established under the uniqueness assumption. Then, a semidiscrete Galerkin method for spatial direction keeping time variable continuous is considered and asymptotic behavior of the semidiscrete solution is derived. Moreover, optimal error estimates are achieved for large time using steady state error estimates. Based on linearized backward Euler method, asymptotic behavior for the fully discrete solution is studied and optimal error estimates are derived for large time. All the results are even valid for κ→0, that is, when the Kelvin-Voigt model converges to the Navier-Stokes system. Finally, some numerical experiments are conducted to confirm our theoretical findings.

Spectral direction splitting methods for two-dimensional space fractional diffusion equations

A numerical method for a kind of time-dependent two-dimensional two-sided space fractional diffusion equations is developed in this paper. The proposed method combines a time scheme based on direction splitting approaches and a spectral method for the spatial discretization. The direction splitting approach renders the underlying two-dimensional equation into a set of one-dimensional space fractional diffusion equations at each time step. Then these one-dimensional equations are solved by using the spectral method based on weak formulations. A time error estimate is derived for the semi-discrete solution, and the unconditional stability of the fully discretized scheme is proved. Some numerical examples are presented to validate the proposed method.

A fast numerical method for the valuation of American lookback put options

A fast and efficient numerical method is proposed and analyzed for the valuation of American lookback options. American lookback option pricing problem is essentially a two-dimensional unbounded nonlinear parabolic problem. We reformulate it into a two-dimensional parabolic linear complementary problem (LCP) on an unbounded domain. The numeraire transformation and domain truncation technique are employed to convert the two-dimensional unbounded LCP into a one-dimensional bounded one. Furthermore, the variational inequality (VI) form corresponding to the one-dimensional bounded LCP is obtained skillfully by some discussions. The resulting bounded VI is discretized by a finite element method. Meanwhile, the stability of the semi-discrete solution and the symmetric positive definiteness of the full-discrete matrix are established for the bounded VI. The discretized VI related to options is solved by a projection and contraction method. Numerical experiments are conducted to test the performance of the proposed method.

Cubic Hermite collocation solution of Kuramoto–Sivashinsky equation

The solution and analysis of Kuramoto–Sivashinsky equation by cubic Hermite collocation method is performed and a bound for maximum norm of the semi-discrete solution is derived by using Lyapunov functional. Error estimates are also obtained for semi-discrete solutions and verified by numerical experiments.

Unified Analysis of Finite Element Methods for Problems with Moving Boundaries

We present a unified analysis of finite element methods for problems with prescribed moving boundaries. In particular, we study an abstract parabolic problem posed on a moving domain with prescribed evolution, discretized in space with a finite element space that is associated with a moving mesh that conforms to the domain at all times. The moving mesh is assumed to evolve smoothly in time, except perhaps at a finite number of remeshing times where the solution is transferred between finite element spaces via a projection. A key result of our analysis is an abstract estimate for the $L^2$-norm of the error between the exact and semidiscrete solutions at a fixed positive time, expressed in terms of the total variation in time of a quantity that measures the difference between the exact solution at time $t$ and its elliptic projection onto the finite element space at time $t$. Specializing the abstract estimate to particular choices of the mesh motion strategy, finite element space, and projector leads to error estimates in terms of the mesh spacing for various semidiscrete schemes. In particular, the estimate can be specialized to conventional arbitrary Lagrangian--Eulerian (ALE) schemes with remeshing as well as schemes based upon universal meshes, where the mesh motion is derived from small deformations of a periodically updated reference subtriangulation of a background mesh that contains the moving domain. We demonstrate such an application by deducing error estimates of optimal order in the mesh spacing for ALE schemes under mild assumptions on the nature of the mesh deformation and the regularity of the exact solution and the moving domain.

The Coupling of RBF and FDM for Solving Higher Order Fractional Partial Differential Equations

In this work, we describe the radial basis functions for solving the time fractional partial differential equations defined by Caputo sense. These problems can be discretized in the time direction based on finite difference scheme and is continuously approximated by using the radial basis functions in the space direction which achieves the semi-discrete solution. Numerical results accuracy the efficiency of the presented method.

Challenges in adjoint-based optimization of a foam EOR process

We apply adjoint-based optimization to a surfactant-alternating gas foam process using a linear foam model introducing gradual changes in gas mobility and a nonlinear foam model giving abrupt changes in gas mobility as function of oil and water saturations and surfactant concentration. For the linear foam model, the objective function is a relatively smooth function of the switching time. For the nonlinear foam model, the objective function exhibits many small-scale fluctuations. As a result, a gradient-based optimization routine could have difficulty finding the optimal switching time. For the nonlinear foam model, extremely small time steps were required in the forward integration to converge to an accurate solution to the semi-discrete (discretized in space, continuous in time) problem. The semi-discrete solution still had strong oscillations in gridblock properties associated with the steep front moving through the reservoir. In addition, an extraordinarily tight tolerance was required in the backward integration to obtain accurate adjoints. We believe the small-scale oscillations in the objective function result from the large oscillations in gridblock properties associated with the front moving through the reservoir. Other EOR processes, including surfactant EOR and near-miscible flooding, have similar sharp changes and may present similar challenges to gradient-based optimization.

Spurious Resonance in SemiDiscrete Methods for the Korteweg--de Vries Equation

A multiple scales analysis of semidiscrete methods for the Korteweg--de Vries equation is conducted. Methods that approximate the spatial derivatives by finite differences with arbitrary order accuracy and the limiting method, the Fourier pseudospectral method, are considered. The analysis reveals that a resonance effect can occur in the semidiscrete solution but not in the solution of the continuous equation. It is shown for the Fourier pseudospectral discretization that resonance can only be caused by aliased modes. The spurious semidiscrete solutions are investigated in numerical experiments and we suggest methods for avoiding spurious resonance.

Numerical solution of fractional telegraph equation by using radial basis functions

In this paper, we implement the radial basis functions for solving a classical type of time-fractional telegraph equation defined by Caputo sense for (1<α≤2). The presented method which is coupled of the radial basis functions and finite difference scheme achieves the semi-discrete solution. We investigate the stability, convergence and theoretical analysis of the scheme which verify the validity of the proposed method. Numerical results show the simplicity and accuracy of the presented method.

Entropy-stable and entropy-dissipative approximations of a fourth-order quantum diffusion equation

Structure-preserving numerical schemes for a nonlinear parabolic fourth-order equation, modeling the electron transport in quantum semiconductors, with periodic boundary conditions are analyzed. First, a two-step backward differentiation formula (BDF) semi-discretization in time is investigated. The scheme preserves the nonnegativity of the solution, is entropy stable and dissipates a modified entropy functional. The existence of a weak semi-discrete solution and, in a particular case, its temporal second-order convergence to the continuous solution is proved. The proofs employ an algebraic relation which implies the G-stability of the two-step BDF. Second, an implicit Euler and $$q$$ q -step BDF discrete variational derivative method are considered. This scheme, which exploits the variational structure of the equation, dissipates the discrete Fisher information (or energy). Numerical experiments show that the discrete (relative) entropies and Fisher information decay even exponentially fast to zero.

Semi-Discrete Analytical Solution of Lumped Mass Finite Element Method for the Longitudinal Vibrations of an Elastic Bar

The objective of this paper is to present semi-discrete analytical method for the longitudinal vibration of an elastic bar. Using lumped mass finite element method, we first obtain a system of second order ordinary differential equations. In terms of some transform technique we obtain the exact solution to the system, i.e. excellently semi-discrete analytical approximation to the longitudinal vibration. An example is given to illustrate the effectiveness of the proposed method.

NUMERICAL ANALYSIS FOR A LOCALLY DAMPED WAVE EQUATION

We consider a semi-discrete finite element formulation with artificial viscosity for the numerical approximation of a problem that models the damped vibrations of a string with fixed ends. The damping coefficient depends on the spatial variable and is effective only in a sub-interval of the domain. For this scheme, the energy of semi-discrete solutions decays exponentially and uniformly with respect to the mesh parameter to zero. We also introduce an implicit in time discretization. Error estimates for the semi-discrete and fully discrete schemes in the energy norm are provided and numerical experiments performed.

Analysis of the Modified Mass Method for the Dynamic Signorini Problem with Coulomb Friction

The aim of the present work is to analyze the modified mass method for the dynamic Signorini problem with Coulomb friction. We prove that the space semidiscrete problem is equivalent to an upper semicontinuous one-sided Lipschitz differential inclusion and is, therefore, well-posed. We derive an energy balance. Next, considering an implicit time-integration scheme, we prove that, under a CFL-type condition on the discretization parameters, the fully discrete problem is well-posed. For a fixed discretization in space, we also prove that the fully discrete solutions converge to the space semidiscrete solution when the time step tends to zero.

Mixed finite element method for the heat diffusion equation in a random medium

We introduce the dual mixed method for the heat evolution equation in a polygonal domain D with a random heat conduction coefficient and heat flux , ⋄ denoting the Wick product. We prove a priori error estimates for the semi-discrete solution of lowest order of the dual mixed method having K-dimensional polynomial chaos expansion of degree N. Due to the re-entrant corner of the polygonal domain D, appropriate refinement rules must be imposed on the family of triangulations in order to recapture convergence of order one in space.

A SPLITTING METHOD FOR THE CAHN–HILLIARD EQUATION WITH INERTIAL TERM

P. Galenko et al. proposed a Cahn–Hilliard model with inertial term in order to model spinodal decomposition caused by deep supercooling in certain glasses. Here we analyze a finite element space semidiscretization of their model, based on a scheme introduced by C. M. Elliott et al. for the Cahn–Hilliard equation. We prove that the semidiscrete solution converges weakly to the continuous solution as the discretization parameter tends to 0. We obtain optimal a priori error estimates in energy norm and related norms, assuming enough regularity on the solution. We also show that the semidiscrete solution converges to an equilibrium as time goes to infinity and we give a simple finite difference version of the scheme.

Energies and distributions of dislocations in stacked pile-ups

To understand the kinematic and thermodynamic effects of representing discrete dislocations as continuous distributions in their slip planes, we consider stacked double ended pile-ups of edge and screw dislocations and compute their distributions and microstructural energies, i.e., the elastic interaction energies of geometrically necessary dislocations (GNDs). In general, three kinds of representations of GNDs are used: discrete, semi-discrete, and, continuous representation. The discrete representations are closest to reality. Therefore, the corresponding solutions are considered exact. In the semi-discrete representation, the discrete dislocations are smeared out into continuous planar distributions within discrete slip planes. The solutions to problems formulated using different descriptions are different. We consider the errors in: dislocation distributions (number of dislocations), and, microstructural energies; when the discrete description is replaced by the semi-discrete one. Asymptotic expressions are derived for: number of dislocations, maximum slip, and, microstructural energy density. They provide a powerful insight into the behavior of the system, and are accurate for a wide range of parameters. Two characteristic lengths emerge from the analysis: the ratio of pile-up length to slip plane spacing (lambda), and, the ratio of slip plane spacing to the Burgers vector. For large enough lambda and large enough number of dislocations, both the discrete and semi-discrete solutions are well-approximated by asymptotic solutions. Results of a comprehensive numerical study are presented.

Finite element methods for semilinear elliptic and parabolic interface problems

The purpose of this paper is to study the finite element methods for second-order semilinear elliptic and parabolic interface problems in two dimensional convex polygonal domains. Optimal order error estimate in the H 1 -norm is proved for the semilinear elliptic interface problem when the grid lines follow the actual interface. An extension to the semilinear parabolic interface problem is considered, and both semidiscrete and fully discrete schemes are discussed. The convergence of the semidiscrete solution to the exact solution is shown to be of order O ( h ) in the L 2 ( 0 , T ; H 1 ( Ω ) ) -norm. Further, a fully discrete scheme based on backward Euler method is analyzed and optimal energy-norm error estimate is established. The interface is assumed to be of arbitrary shape but is smooth.

Uniform convergence for a finite-element discretization of a viscous diffusion equation

We study a space semidiscretization of a viscous diffusion equation, obtained as a singular limit of the viscous Cahn-Hilliard equation by letting the interfacial energy tend to 0. The semidiscrete solution is shown to converge uniformly in time and space to the continuous solution, on finite time intervals, as the discretization parameter h tends to 0 (in space dimension one, two and three). We obtain an optimal error bound in space dimension one assuming only a piecewise Lipschitz regularity on the initial value. This approach allows us to obtain some counterexamples concerning lower and upper bounds of solutions to Cahn-Hilliard equations. Numerical simulations confirm the theoretical results.

Convergence of a space semi-discrete modified mass method for the dynamic Signorini problem

A new space semi-discretization for the dynamic Signorini problem, based on a modification of the mass term, has been recently proposed. We prove the convergence of the space semi-discrete solutions to a solution of the continuous problem in the case of a visco-elastic material.

Convergence of an entropic semi-discretization for nonlinear Fokker-Planck equations in $\mathbb{R}^d$

A nonlinear degenerate Fokker-Planck equation in the whole space is analyzed. The existence of solutions to the corresponding implicit Euler scheme is proved, and it is shown that the semi-discrete solution converges to a solution of the continuous problem. Furthermore, the discrete entropy decays monotonically in time and the solution to the continuous problem is unique. The nonlinearity is assumed to be of porous-medium type. For the (given) potential, either a less than quadratic growth condition at infinity is supposed or the initial datum is assumed to be compactly supported. The existence proof is based on regularization and maximum principle arguments. Upper bounds for the tail behavior in space at infinity are also derived in the at-most-quadratic growth case.