Semi-discrete Problem Research Articles

An a priori error analysis for a projection based variational multiscale finite element method for Oseen problems in a time-dependent domain

Stability and error estimates for a projection based variational multiscale finite element scheme for Oseen problem in a time-dependent domain are derived in this paper. The use of Geometric Conservation Law (GCL) provides an unconditional stable scheme, whereas a restriction on the time-step needs to be imposed to obtain stability estimates independent of the mesh velocity when GCL is violated. Further, a priori error estimate is derived for the semi-discrete problem obtained with the backward Euler time discretization.

A Numerical Model for Steel Continuous Casting Problem in a Time-variable Domain

A mathematical model and numerical method for simulation of the continuous casting process in a variable in time domain are presented. The variable geometry of the slab is caused by the change in time of the width of the mould. The mathematical model of the process is a Stefan problem with prescribed convection and non-linear Robin boundary condition. Considered differential equation is approximated by a finite difference scheme, which is constructed in several steps. First, a semi-discrete problem is constructed using the method of characteristics with respect to the time variable. Then, at each time level, the current elliptic problem in a curvilinear domain is replaced by a problem in the parallelepiped domain using the fictitious domain method. Finally, the boundary-value problem in the parallelepiped domain is approximated by a finite-difference scheme. The constructed fully discrete problem in algebraic form is a system of nonlinear equations containing a diagonal monotone operator and a linear part with a symmetric and positive definite $$M$$ -matrix. To solve the resulting system of nonlinear algebraic equations, well-known iterative solution methods can be applied.

Spectral discretization of time-dependent vorticity\u2013velocity\u2013pressure formulation of the Navier\u2013Stokes equations

In this work, we propose a nonstationary Navier–Stokes problem equipped with an unusual boundary condition. The time discretization of such a problem is based on the backward Euler’s scheme. However, the variational formulation deduced from the nonstationary Navier–Stokes equations is discretized using the spectral method. We prove that the time semidiscrete problem and the full spectral discrete one admit at most one solution.

Transparent boundary conditions for wave propagation in fractal trees: convolution quadrature approach

In this work we propose high-order transparent boundary conditions for the weighted wave equation on a fractal tree, with an application to the modeling of sound propagation in a human lung. This article follows the recent work (Joly et al. in Netw Heterog Media 14(2):205–264, 2019), dedicated to the mathematical analysis of the corresponding problem and the construction of low-order absorbing boundary conditions. The method proposed in this article consists in constructing the exact (transparent) boundary conditions for the semi-discretized problem, in the spirit of the convolution quadrature method developed by Ch. Lubich. We analyze the stability and convergence of the method, and propose an efficient algorithm for its implementation. The exposition is concluded with numerical experiments.

Boundary update via resolvent for fluid-structure interaction

Abstract We propose a BOundary Update using Resolvent (BOUR) partitioned method, second-order accurate in time, unconditionally stable, for the interaction between a viscous, incompressible fluid and a thin structure. The method is algorithmically similar to the sequential Backward Euler - Forward Euler implementation of the midpoint quadrature rule. (i) The structure and fluid sub-problems are first solved using a Backward Euler scheme, (ii) the velocities of fluid and structure are updated on the boundary via a second-order consistent resolvent operator, and then (iii) the structure and fluid sub-problems are solved again, using a Forward Euler scheme. The stability analysis based on energy estimates shows that the scheme is unconditionally stable. Error analysis of the semi-discrete problem yields second-order convergence in time. The two numerical examples confirm theoretical convergence analysis results and show an excellent agreement between the proposed partitioned scheme and the monolithic scheme.

Mixed FEM for Time-Fractional Diffusion Problems with Time-Dependent Coefficients

In this paper, a mixed finite element method is applied in spatial directions while keeping time variable continuous to a class of time-fractional diffusion problems with time-dependent coefficients on a bounded convex polygonal domain. Based on an energy argument combined with a repeated application of an integral operator, optimal error estimates, which are optimal with respect to both approximation properties and regularity results, are derived for the semidiscrete problem with smooth as well as nonsmooth initial data. Specially, a priori error bounds for both primary and secondary variables in $$L^2$$-norm are established. Since the comparison between Fortin projection and the mixed Galerkin approximation of the secondary variable yields an improved rate of convergence, therefore, as a by-product, we derive $$L^p$$-estimates for the error in primary variable. Finally, some numerical experiments are conducted to confirm our theoretical findings.

A three-dimensional enriched finite element method for nonlinear transient heat transfer in functionally graded materials

Nonlinear transient heat transfer in functionally graded materials is being studied more popular in present. In preliminary design, this problem can be simplified as a composite, and a three-dimensional transient heat transfer analysis is used to adjust dimensions of the considered materials. This paper is concerned with the numerical modeling of transient heat transfer in composite materials where the thermal conductivity is also dependent on the temperature; hence the problem is nonlinear. We are interested in solutions with steep boundary layers where highly refined meshes are commonly needed. Such problems can be challenging to solve with the conventional finite element method. To deal with this challenge we propose an enriched finite element formulation where the basis functions are augmented with a summation of exponential functions. First, the initial-value problem is integrated in time using a semi-implicit scheme and the semi-discrete problem is then integrated in space using the enriched finite elements. We demonstrate through several numerical examples that the proposed approach can recover the heat transfer on coarse meshes and with much fewer degrees of freedom compared to the standard finite element method. Thus, a significant reduction in the computational requirements is achieved without compromising on the solution accuracy. The results also show the stability of the scheme when using tetrahedral unstructured grids.

Strong convergence of a stochastic Rosenbrock-type scheme for the finite element discretization of semilinear SPDEs driven by multiplicative and additive noise

This paper aims to investigate the numerical approximation of a general second order parabolic stochastic partial differential equation(SPDE) driven by multiplicative and additive noise. Our main interest is on such SPDEs where the nonlinear part is stronger than the linear part, usually called stochastic dominated transport equations. Most standard numerical schemes lose their good stability properties on such equations, including the current linear implicit Euler method. We discretize the SPDE in space by the finite element method and propose a novel scheme called stochastic Rosenbrock-type scheme for temporal discretization. Our scheme is based on the local linearization of the semi-discrete problem obtained after space discretization and is more appropriate for such equations. We provide a strong convergence of the new fully discrete scheme toward the exact solution for multiplicative and additive noise and obtain optimal rates of convergence. Numerical experiments to sustain our theoretical results are provided.

Analytical Insights into a Generalized Semidiscrete System with Time-Varying Coefficients: Derivation, Exact Solutions, and Nonlinear Soliton Dynamics

In this paper, a new generalized semidiscrete integrable system with time-varying coefficients is analytically studied. Firstly, the generalized semidiscrete system is derived from a semidiscrete matrix spectral problem by embedding finite time-varying coefficient functions. Secondly, exact and explicit N-soliton solutions of the semidiscrete system are obtained by using the inverse scattering analysis. Finally, three special cases when N=1,2,3 of the obtained N-soliton solutions are simulated by selecting some appropriate coefficient functions. It is shown that the time-varying coefficient functions affect the spatiotemporal structures and the propagation velocities of the obtained semidiscrete one-soliton solutions, two-soliton solutions, and three-soliton solutions.

Partition of Unity Finite Element Analysis of Nonlinear Transient Diffusion Problems Using p-Version Refinement

We propose a high-order enriched partition of unity finite element method for linear and nonlinear time-dependent diffusion problems. The solution of this class of problems often exhibits non-smooth features such as steep gradients and boundary layers which can be very challenging to recover using the conventional low-order finite element methods. A class of steady-state exponential functions has been widely used for enrichment and its performance to numerically solve these challenges has been demonstrated. However, these enrichment functions have been used only in context of the standard h-version refinement or the so-called q-version refinement. In this paper we demonstrate that the p-version refinement can also be a very attractive option in terms of the efficiency and the accuracy in the enriched partition of unity finite element method. First, the transient diffusion problem is integrated in time using a semi-implicit scheme and the semi-discrete problem is then integrated in space using the p-version enriched finite elements. Numerical results are presented for three test examples of timedependent diffusion problems in both homogeneous and heterogeneous media. The computed results show the significant improvement when using the p-version refined enriched approximations in the finite element analysis. In addition, these results support our expectations for a robust and high-order accurate enriched partition of unity finite element method.

Existence Results for Fractional Diffusion Equation with Dirichlet Boundary Condition

In this article, we investigate the time-fractional diffusion equation with homogeneous Dirichlet boundary condition. The existence and uniqueness of a solution to the equation is proved using Lax-Milgram theorem. Then the time is discretized with the help of the finite difference scheme and stability result for the semi-discretized problem is shown. Furthermore, space is discretized using the idea of triangulation in the finite element method.

A robust adaptive grid method for singularly perturbed Burger-Huxley equations

<p style='text-indent:20px;'>In this paper, an adaptive grid method is proposed to solve one-dimensional unsteady singularly perturbed Burger-Huxley equation with appropriate initial and boundary conditions. Firstly, we use the classical backward-Euler scheme on a uniform mesh to approximate time derivative. The resulting nonlinear singularly perturbed semi-discrete problem is linearized by using Newton-Raphson-Kantorovich approximation method which is quadratically convergent. Then, an upwind finite difference scheme on an adaptive nonuniform grid is used for space derivative. The nonuniform grid is generated by equidistribution of a positive monitor function, which is similar to the arc-length function. It is shown that the presented adaptive grid method is first order uniform convergent in the time and spatial directions, respectively. Finally, numerical results are given to validate the theoretical results.</p>

Finite element solution of isothermal gas flow in a network

In a previous paper, [1], a new formulation for isothermal gas flow in a single pipeline including gravity and friction effects has been introduced and solved by finite element methods. A performance analysis has been done by using some tests both manufactured and experimental. The goal of the present work is to extend the ideas to model gas flow in a transportation network. As a first step, in this paper the network only includes pipes connected at nodes which can be emission or consumption points, or simply structural connections between pipes. In addition to the mass and momentum conservation equations inside the pipes that are partial differential equations, we need to write the mass conservation equations at the nodes. The latter appear to be a set of constraints (one per node) whose respective Lagrange multipliers are the (continuous) pressures at the nodes. Thus, the whole model is a Partial Differential-Algebraic Equation (PDAE). For numerical solution, an implicit time discretization is proposed first and then a weak formulation of the resulting semi-discrete problem is introduced. After doing a finite element discretization we are led to solve a nonlinear system of numerical equations per time step what is done by Newton's method. In order to reduce the computing time, the algorithm can be parallelized in such a way that, at each time step, every pipe can be solved separately in one processor (or core). Moreover, we show that the proposed scheme exactly conserves the mass at the nodes and in the whole network, and a new simple and compact way of computing the network line-pack over the time is given. Finally, the introduced methodology is validated by applying the computer program to a triangular network, which is a standard test in references on the subject, and to a real network for which we have got measurements that enable us to create the boundary conditions for the model and also to compare the numerical results with experimental data.

Conservative Local Discontinuous Galerkin method for the fractional Klein-Gordon-Schrödinger system with generalized Yukawa interaction

The formulation of the Local Discontinuous Galerkin (LDG) method applied to the space fractional Klein-Gordon-Schrodinger system with generalized interaction is presented. By considering its primal formulation and taking advantage of the symmetry of the bilinear form associated to the discretization of the Riesz differential operator, conservation of discrete analogues of the mass and the energy can be demonstrated for the semi-discrete problem and for the fully discrete problem using, as time marching scheme, a combination of the modified Crank-Nicolson method for the fractional nonlinear Schrodinger equation and the Newmark method for the nonlinear Klein-Gordon equation. Boundedness of the numerical solution in the L2 norm is derived from the conservation properties of the fully discrete method. A series of numerical experiments with high order approximations illustrates our conservation results and shows that optimal rates of convergence can be also achieved.

A fast compact time integrator method for a family of general order semilinear evolution equations

In this paper we develop a fast compact time integrator method for numerically solving a family of general order semilinear evolution equations in regular domains. The spatial discretization is carried out by a fourth-order accurate compact difference scheme in which fast Fourier transform can be utilized for efficient implementation. The resulting semi-discretized problem consists of a system of ordinary differential equations whose solution can be explicitly expressed in term of time integrators, and a desired numerical method is then obtained by further adopting multistep approximations of the nonlinear terms based on the solution formula. Linear stability analysis is performed for the method for second-order in time evolution equations. Extensive numerical experiments with applications are also presented to demonstrate efficiency, accuracy, and stability of the proposed method in practice.

A thermodynamically consistent model for convective transport in nanofluids: existence of weak solutions and fem computations

We present a mathematical model for convective transport in nanofluids including thermophoretic effects that is very similar to the celebrated model of Buongiorno [4]. Our model is thermodynamically consistent and consequently an energy estimate can be shown. We propose a semi–discretization in time that fully decouples the subproblems. Also for this semi–discrete problem an energy estimate can rigorously be shown. Based on this energy estimate it is proved that solutions of the semi–discrete problem converge to a weak solution of the system. We use the time discretization to define an effective, fully discrete finite element scheme. Simulations are performed for a nanofluid flowing through a heated long pipe. Careful inspection of the computational results reveal the mechanism of enhanced cooling properties of the nanofluid compared to the base fluid: The temperature gradient at the wall reduces the concentration of particles by thermophoresis. Thus, close to the wall, the viscosity is smaller compared to the bulk leading to an enhanced convective transport in the boundary layer.

An HDG Method for the Time-dependent Drift–Diffusion Model of Semiconductor Devices

We propose a hybridizable discontinuous Galerkin (HDG) finite element method to approximate the solution of the time dependent drift-diffusion problem. This system involves a nonlinear convection diffusion equation for the electron concentration $u$ coupled to a linear Poisson problem for the the electric potential $\phi$. The non-linearity in this system is the product of the $\nabla \phi$ with $u$. An improper choice of a numerical scheme can reduce the convergence rate. To obtain optimal HDG error estimates for $\phi$, $u$ and their gradients, we utilize two different HDG schemes to discretize the nonlinear convection diffusion equation and the Poisson equation. We prove optimal order error estimates for the semidiscrete problem. We also present numerical experiments to support our theoretical results.

A Priori Error Analysis for Navier Stokes Equations with Slip Boundary Conditions of Friction Type

The time dependent Navier Stokes equations under nonlinear slip boundary conditions are discretized by backward Euler scheme in time and finite elements in space. We derive error estimates for the semi-discrete problems. The focus on the semi discrete problem in time is to obtain convergence rate without extra regularity on the weak solution by following Nochetto et al. (Commun Pure Appl Math 53(5):525–589, 2000). The semi discrete problem in space is analyzed with the help of the Stokes operator introduced. Finally we use the triangle inequality to derive the global a priori error estimates.

Numerical Quenching for Heat Equations with Coupled Nonlinear Boundary Flux

In this paper, we study a numerical approximation of the following problem ut = uxx, vt = vxx, 0 < x < 1, 0 < t < T; ux(0, t) = u−m(0, t) + v−p(0, t), vx(0, t) = u−q (0, t) + v−n(0, t) and ux(1, t) = vx(1, t) = 0, 0 < t < T, where m, p, q and n are parameters. We prove that the solution of a semidiscrete form of above problem quenches in a finite time only at first node of the mesh. We show that the time derivative of the solution blows up at quenching node. Some conditions under which the non-simultaneous or simultaneous quenching occurs for the solution of the semidiscrete problem are obtained. We establish the convergence of the quenching time. Finally, some numerical results to illustrate our analysis are given.

Towards the modeling of the Purkinje/myocardium coupled problem: A well-posedness analysis

The Purkinje network is the specialized conduction system in the heart. It ensures the physiological spread of the electrical wave in the ventricles. In this work, in an insulated heart framework, we model the free running Purkinje system, using the monodomain equation. The intra-myocardium part of the Purkinje fiber is coupled to the ventricular tissue using the bidomain equation. The coupling is performed through the extracellular potential. We discretize the problem in time using a semi-implicit scheme. Then, we write a variational formulation of the semi discrete problem in a non standard weighted Sobolev functional spaces. We prove the existence and uniqueness of the solution of the Purkinje/myocardium semi-discretized problem. We discretize in space by the finite element P 1 − L a g r a n g e and conduct some numerical tests showing the anterograde and retrograde propagation of the electrical wave between the tissue and the Purkinje fibers.