Semidiscrete Approximation Research Articles

A mass-lumped mixed finite element method for acoustic wave propagation

We consider the numerical approximation of acoustic wave propagation in time domain by a mixed finite element method based on the $$\text {BDM}_1$$–$$\text {P}_0$$ spaces. A mass-lumping strategy for the $$\text {BDM}_1$$ element, originally proposed by Wheeler and Yotov in the context of subsurface flow, is utilized to enable an efficient integration in time. By this mass-lumping strategy, the accuracy of the mixed method is formally reduced to first order. We will show however that the numerical approximation still carries global second order information which is expressed in the super-convergence of the numerical approximation to certain projections of the true solution. Based on this fact, we propose post-processing strategies for the pressure and the velocity leading to piecewise linear approximations with second order accuracy. A complete convergence analysis is provided for the semi-discrete and corresponding fully-discrete approximations, which result from time discretization by the leapfrog method. The efficiency of the proposed strategy is illustrated also in numerical tests.

Semi-discrete finite-element approximation of nonlocal hyperbolic problem

ABSTRACT In this paper, we investigate a semi-discrete finite-element approximation of nonlocal hyperbolic problem. A priori error estimate for the semi-discrete scheme is derived. A fully discrete scheme based on backward difference method is constructed. We discuss the existence-uniqueness of the solution for fully discrete problem. In order to linearize the nonlinear fully discrete problem, we use Newton's method. Numerical results based on the usual finite-element method are provided to confirm the theoretical estimate.

A new class of hyperbolic variational–hemivariational inequalities driven by non-linear evolution equations

The aim of the paper is to introduce and investigate a dynamical system which consists of a variational–hemivariational inequality of hyperbolic type combined with a non-linear evolution equation. Such a dynamical system arises in studies of complicated contact problems in mechanics. Existence, uniqueness and regularity of a global solution to the system are established. The approach is based on a new semi-discrete approximation with an application of a surjectivity result for a pseudomonotone perturbation of a maximal monotone operator. A new dynamic viscoelastic frictional contact model with adhesion is studied as an application, in which the contact boundary condition is described by a generalised normal damped response condition with unilateral constraint and a multivalued frictional contact law.

Spatial convergence for semi-linear backward stochastic differential equations in Hilbert space: a mild approach

In this paper, we present convergence results of a spatial semi-discrete approximation of a Hilbert space-valued backward stochastic differential equations with noise driven by a cylindrical Q-Wiener process. Both the solution and its space discretization are formulated in mild forms. Under suitable assumptions of the final value and the drift, a convergence rate is established.

Stable Symmetric Matrix Form Framework for the Elastic Wave Equation Combined with Perfectly Matched Layer and Discretized in the Curve Domain

In this paper, we present a stable and accurate high-order methodology for the symmetric matrix form (SMF) of the elastic wave equation. We use an accurate high-order upwind finite difference method to define spatial discretization. Then, an efficient complex frequency-shifted (CFS) unsplit multi-axis perfectly matched layer (MPML) is implemented using the auxiliary differential equation (ADE) that is used to build higher-order time schemes for elastodynamics in the unbounded curve domain. It is derived to be compatible with SMF. The SMF framework has a general form of a hyperbolic partial differential equation (PDE) that can be expanded to different dimensions (2D, 3D) or different wave modal (SH, P-SV) without requiring significant modifications owing to a simplified process of derivation and programming. Subsequently, an energy analysis on the framework combined with initial boundary value problems is conducted, and the stability analysis can be extended to a semi-discrete approximation similarly. Thus, we propose a semi-discrete approximation based on ADE CFS-MPML in which the curve domain is discretized using the upwind summation-by-parts (SBP) operators, and where the boundary conditions are enforced weakly using the simultaneous approximation terms (SAT). The proposed method’s robustness and adequacy are illustrated by conducting several numerical simulations.

Thermodynamically consistent spatial discretization of the one-dimensional regularized system of the Navier–Stokes–Cahn–Hilliard equations

This paper is devoted to the construction of a thermodynamically consistent (with non-increasing total energy on the discrete level) semi-discrete approximation (continuous in time and discrete in space) for the one-dimensional regularized system of the Navier–Stokes–Cahn–Hilliard equations that describes a compressible isothermal flow of two-phase binary fluid with surface effects. Thermodynamical consistency is achieved using special form of momentum equation with surface force term written in so called potential form. Results of numerical study are presented.

Absorbing boundary conditions for the time-dependent Schrödinger-type equations inR3

Absorbing boundary conditions are presented for three-dimensional time-dependent Schrödinger-type of equations as a means to reduce the cost of the quantum-mechanical calculations. The boundary condition is first derived from a semidiscrete approximation of the Schrödinger equation with the advantage that the resulting formulas are automatically compatible with the finite-difference scheme and no further discretization is needed in space. The absorbing boundary condition is expressed as a discrete Dirichlet-to-Neumann map, which can be further approximated in time by using rational approximations of the Laplace transform to enable a more efficient implementation. This approach can be applied to domains with arbitrary geometry. The stability of the zeroth-order and first-order absorbing boundary conditions is proved. We tested the boundary conditions on benchmark problems. The effectiveness is further verified by a time-dependent Hartree-Fock model with Skyrme interactions. The accuracy in terms of energy and nucleon density is examined as well.

LQR Control of Moisture Distribution in Microwave Drying Process Based on a Finite Element Model of Parabolic PDEs

Abstract The microwave drying process is a widely used technology in the drying of porous dielectric materials. Designing a controller for moisture distribution in this process can improve product quality and reduce energy consumption and production time. In this paper, a model-based controller for moisture distribution in an industrial microwave drying process is developed. The moisture and temperature in this process are described by a pair of partial differential equations (PDEs) and have both temporal and spatial variations. In this view, using a semi-discrete finite element approximation, the coupled system of PDEs is transformed into a system of ordinary differential equations (ODEs). Based on the discretized ODEs, a linear quadratic regulator (LQR) controller is designed to determine the power levels of multiple microwave sources in this process to reach and maintain the desired moisture level. Numerical simulations are carried out in three different drying scenarios. The results show that the proposed controller achieves a very good performance in tracking the desired moisture level.

A Posteriori Error Estimation for the p-Curl Problem

We derive a posteriori error estimates for a semidiscrete finite element approximation of a nonlinear eddy current problem arising from applied superconductivity, known as the p-curl problem. In particular, we show the reliability for nonconforming Nédélec elements based on a residual-type argument and a Helmholtz--Weyl decomposition of W^p_0(curl;Omega). As a consequence, we are also able to derive an a posteriori error estimate for a quantity of interest called the AC loss. The nonlinearity for this form of Maxwell's equation is an analogue of the one found in the p-Laplacian. It is handled without linearizing around the approximate solution. The nonconformity is dealt with by adapting error decomposition techniques of Carstensen, Hu, and Orlando. Geometric nonconformities also appear because the continuous problem is defined over a bounded C^1,1 domain, while the discrete problem is formulated over a weaker polyhedral domain. The semidiscrete formulation studied in this paper is often encountered in commercial codes and is shown to be well-posed. The paper concludes with numerical results confirming the reliability of the a posteriori error estimate.

Finite element methods for non-fourier thermal wave model of bio heat transfer with an interface

We propose a fitted finite element method for non-Fourier bio heat transfer model in multi-layered media. Specifically, we employ the Maxwell–Cattaneo equation on the physical media that have a heterogeneous conductivity. Well-posedness of the model interface problem is established. A continuous piecewise linear finite element space is employed for the spatially semidiscrete approximation and the temporal discretization is based on backward scheme. Optimal order error estimates for both semidiscrete and fully discrete schemes are proved in $$L^{\infty }(H^1)$$ norm. Finally, we give numerical examples to verify our theoretical results. The new results and finite element schemes can be applied in the fields of engineering, medicine, and biotechnology.

Modulational instability in addition to discrete breathers in 2D quantum ultracold atoms loaded in optical lattices

The modulational instability associated with discrete breathers in 2D quantum ultracold atoms is studied by using the Glauber’s coherent state combined with a semi-discrete approximation and multiple-scale methods. The linear stability analysis exhibits an intriguing threshold amplitude and instability regions associated with modulational growth rate. In addition, we demonstrate a coexistence of two bright intrinsic localized modes namely, the radial symmetric and bilateral symmetric modes, at the center and at the edges of the Brillouin zone, respectively, by alternating the on-site parameter interaction. Numerical investigations reveal a good agreement with the theoretical analysis.

Finite Element Convergence for State-Based Peridynamic Fracture Models

We establish the a priori convergence rate for finite element approximations of a class of nonlocal nonlinear fracture models. We consider state-based peridynamic models where the force at a material point is due to both the strain between two points and the change in volume inside the domain of the nonlocal interaction. The pairwise interactions between points are mediated by a bond potential of multi-well type while multi-point interactions are associated with the volume change mediated by a hydrostatic strain potential. The hydrostatic potential can either be a quadratic function, delivering a linear force–strain relation, or a multi-well type that can be associated with the material degradation and cavitation. We first show the well-posedness of the peridynamic formulation and that peridynamic evolutions exist in the Sobolev space $$H^2$$. We show that the finite element approximations converge to the $$H^2$$ solutions uniformly as measured in the mean square norm. For linear continuous finite elements, the convergence rate is shown to be $$C_t \Delta t + C_s h^2/\epsilon ^2$$, where $$\epsilon $$ is the size of the horizon, h is the mesh size, and $$\Delta t$$ is the size of the time step. The constants $$C_t$$ and $$C_s$$ are independent of $$\Delta t$$ and h and may depend on $$\epsilon $$ through the norm of the exact solution. We demonstrate the stability of the semi-discrete approximation. The stability of the fully discrete approximation is shown for the linearized peridynamic force. We present numerical simulations with the dynamic crack propagation that support the theoretical convergence rate.

Analyticity, Regularity, and Generalized Polynomial Chaos Approximation of Stochastic, Parametric Parabolic Two-Scale Partial Differential Equations

We study two-scale parabolic partial differential equations whose coefficient is stochastic and depends linearly on a sequence of pairwise independent random variables which are uniformly distributed in a compact interval. We cast the problem into a deterministic two-scale parabolic problem which depends on a sequence of real parameters in a compact interval. Passing to the limit when the microscale tends to zero, using two-scale homogenization, we obtain the parametric two-scale homogenized problem. This problem contains the solution to the homogenized equation which describes the solution to the original two-scale parabolic problem macroscopically, and the corrector which encodes the microscopic information. The solution of this two-scale homogenized equation is represented as a generalized polynomial chaos (gpc) expansion according to a polynomial basis of the L2 space of the parameters. We use a semidiscrete Galerkin approximation which projects the solution into a space of parametric functions which contain a finite number of pre-chosen gpc modes. Analyticity of the solution of the two-scale homogenized equation with respect to the parameters is established. Under mild assumptions, we show the summability of the coefficients of the solution’s gpc expansion. From this, an explicit error estimate for the semidiscrete Galerkin approximation in terms of the number of the chosen gpc modes is derived when these gpc modes are chosen as the N best ones according to the norms of the gpc coefficients. Regularity of the gpc coefficients and summability of their norms in the regularity spaces are also established. Using the solution of the best N term semidiscrete Galerkin approximation, we derive an approximation for the solution of the original two-scale problem, with an explicit convergence rate in terms of the microscopic scale and the number of gpc modes in the Galerkin approximation.

On the convergence rates of energy-stable finite-difference schemes

Abstract We consider constant-coefficient initial-boundary value problems, with a first or second derivative in time and a highest spatial derivative of order q, and their semi-discrete finite difference approximations. With an internal truncation error of order p ≥ 1 , and a boundary error of order r ≥ 0 , we prove that the convergence rate is: min ⁡ ( p , r + q ) . The assumptions needed for these results to hold are: i) The continuous problem is linear and well-posed (with a smooth solution). ii) The numerical scheme is consistent, nullspace consistent, nullspace invariant, and energy stable. These assumptions are often satisfied for Summation-By-Parts schemes.

Oscillating two-dimensional Ca2+ waves in cell networks with bidirectional paracrine signaling

Nonlinear excitations of waves are investigated in a two-dimensional cell network with bidirectional paracrine signaling, both in the longitudinal and transversal directions. The semi-discrete approximation is used to show that the dynamics of the intercellular waves can be reduced to complex Ginzburg–Landau equations, depending on the high- or low-frequency regime. The onset of modulational instability is addressed, where the instability features of the low- and high-frequency modes are compared via the instability growth rate. The -expansion method is employed to find analytically spiral-like wave solutions for the two dynamical regimes. Their response to the effect of paracrine coupling is also addressed.

Local Discontinuous Galerkin Scheme for Space Fractional Allen–Cahn Equation

This paper is concerned with the efficient numerical solution for a space fractional Allen–Cahn (AC) equation. Based on the features of the fractional derivative, we design and analyze a semi-discrete local discontinuous Galerkin (LDG) scheme for the initial-boundary problem of the space fractional AC equation. We prove the optimal convergence rates of the semi-discrete LDG approximation for smooth solutions. Finally, we test the accuracy and efficiency of the designed numerical scheme on a uniform grid by three examples. Numerical simulations show that the space fractional AC equation displays abundant dynamical behaviors.

Galerkin finite element methods for the Shallow Water equations over variable bottom

We consider the one-dimensional shallow water equations (SW) in a finite channel with variable bottom topography. We pose several initial-boundary-value problems for the SW system, including problems with transparent (characteristic) boundary conditions in the supercritical and the subcritical case. We discretize these problems in the spatial variable by standard Galerkin-finite element methods and prove L^2-error estimates for the resulting semidiscrete approximations. We couple the schemes with the 4th order-accurate, explicit, classical Runge-Kutta time stepping procedure and use the resulting fully discrete methods in numerical experiments of shallow water wave propagation over variable bottom topographies with several kinds of boundary conditions. We discuss issues related to the attainment of a steady state of the simulated flows, including the good balance of the schemes.

Localized nonlinear excitations in diffusive memristor-based neuronal networks

We extend the existing ordinary differential equations modeling neural electrical activity to include the memory effect of electromagnetic induction through magnetic flux, used to describe time varying electromagnetic field. Through the multi-scale expansion in the semi-discrete approximation, we show that the neural network dynamical equations can be governed by the complex Ginzburg-Landau equation. The analytical and numerical envelop soliton of this equation are reported. The results obtained suggest the possibility of collective information processing and sharing in the nervous system, operating in both the spatial and temporal domains in the form of localized modulated waves. The effects of memristive synaptic electromagnetic induction coupling and perturbation on the modulated action potential dynamics examined. Large electromagnetic induction coupling strength may contribute to signal block as the amplitude of modulated waves are observed to decrease. This could help in the development of a chemical brain anaesthesia for some brain pathologies.

The Conservative Time High-Order AVF Compact Finite Difference Schemes for Two-Dimensional Variable Coefficient Acoustic Wave Equations

In this paper, we develop and analyze the energy conservative time high-order AVF compact finite difference methods for variable coefficient acoustic wave equations in two dimensions. We first derive out an infinite-dimensional Hamiltonian system for the variable coefficient wave equations and apply the spatial fourth-order compact finite difference operator to the equations of the system to obtain a semi-discrete approximation system, which can be cast into a canonical finite-dimensional Hamiltonian form. We then apply the second-order and fourth-order AVF techniques to propose the fully discrete energy conservative time high-order AVF compact finite difference methods for wave equations in two dimensions. We prove that the proposed semi-discrete and fully-discrete schemes satisfy energy conservations in the discrete forms. We further prove that the semi-discrete scheme has the fourth-order convergence order in space and the fully-discrete AVF compact finite difference method has the fourth-order convergence order in both time and space. Numerical tests confirm the theoretical results.

Error estimates for Galerkin finite element methods for the Camassa–Holm equation

We consider the Camassa–Holm (CH) equation, a nonlinear dispersive wave equation that models one-way propagation of long waves of moderately small amplitude. We discretize in space the periodic initial-value problem for CH (written in its original and in system form), using the standard Galerkin finite element method with smooth splines on a uniform mesh, and prove optimal-order $$L^{2}$$ -error estimates for the semidiscrete approximation. Using the fourth-order accurate, explicit, “classical” Runge–Kutta scheme for time-stepping, we construct a highly accurate, stable, fully discrete scheme that we employ in numerical experiments to approximate solutions of CH, mainly smooth travelling waves and nonsmooth solitons of the ‘peakon’ type.