Semidiscrete Finite Element Approximation Research Articles

Finite element analysis of extended Fisher-Kolmogorov equation with Neumann boundary conditions

This paper delves into the numerical analysis of the extended Fisher-Kolmogorov (EFK) equation within open bounded convex domains R⊂Rd, where d≤3. Two distinct finite element schemes are introduced, namely the semi-discrete and fully-discrete finite element approximations. The existence and uniqueness of solutions are established for both the semi-discrete and fully-discrete finite element approximations. Error bounds are investigated across different scenarios, including comparisons between the semi-discrete and exact solutions, as well as between the semi-discrete and fully-discrete solutions, along with the fully-discrete and exact solutions. An effective algorithm has been proposed to solve the nonlinear system resulting from the fully-discrete finite element approximation at each time step. The paper also provides computed numerical error results and showcases a variety of numerical experiments to further illustrate and support the findings.

Error estimates of semidiscrete finite element approximation for minimal time impulse control problems

Error estimates of semidiscrete finite element approximation for minimal time impulse control problems

A novel finite element approximation of anisotropic curve shortening flow

We extend the DeTurck trick from the classical isotropic curve shortening flow to the anisotropic setting. Here, the anisotropic energy density is allowed to depend on space, which allows an interpretation in the context of Finsler metrics, giving rise to, for instance, geodesic curvature flow in Riemannian manifolds. Assuming that the density is strictly convex and smooth, we introduce a novel weak formulation for anisotropic curve shortening flow. We then derive an optimal H^1 -error bound for a continuous-in-time semidiscrete finite element approximation that uses piecewise linear elements. In addition, we consider some fully practical fully discrete schemes and prove their unconditional stability. Finally, we present several numerical simulations, including some convergence experiments that confirm the derived error bound, as well as applications to crystalline curvature flow and geodesic curvature flow.

Numerical analysis for a system coupling curve evolution attached orthogonally to a fixed boundary, to a reaction–diffusion equation on the curve

AbstractWe consider a semidiscrete finite element approximation for a system consisting of the evolution of a planar curve evolving by forced curve shortening flow inside a given bounded domain , such that the curve meets the boundary orthogonally, and the forcing is a function of the solution of a reaction–diffusion equation that holds on the evolving curve. We prove optimal order error bounds for the resulting approximation and present numerical experiments.

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.

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

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 approximations of impulsive optimal control problems for heat equations

In this paper we study the priori error analysis for semidiscrete Galerkin finite element approximation of the impulsive optimal control problem governed by linear heat equation with control constraint. We first derive Pontryagin's maximum principle and construct the semidiscrete finite element approximation scheme for our impulsive optimal control problem. Then, the error estimates for optimal control u, the related state y and the adjoint state φ will be obtained. Finally, an efficient algorithm for our numerical experiment is designed, which verifies the theoretical results obtained in this paper.

Optimal convergence rates for semidiscrete finite element approximations of linear space-fractional partial differential equations under minimal regularity assumptions

We consider the optimal convergence rates of the semidiscrete finite element approximations for solving linear space-fractional partial differential equations by using the regularity results for the fractional elliptic problems obtained recently by Jin et al. (2015) and Ervin et al. (2018). The error estimates are proved by using two approaches. One approach is to apply the duality argument in Johnson (1987) for the heat equation to consider the error estimates for the linear space-fractional partial differential equations. This argument allows us to obtain the optimal convergence rates under the minimal regularity assumptions for the solution. Another approach is to use the approximate solution operators of the corresponding fractional elliptic problems. This argument can be extended to consider more general linear space-fractional partial differential equations. Numerical examples are given to show that the numerical results are consistent with the theoretical results.

Wave propagation in initially-stressed elastic rods

This paper presents a study of incremental wave motion in elastic media with initial stress. Alternative formulations of the governing equations are examined as independent developments as well as special cases of a general theory in which the strain energy function is written in terms of deformation and initial stress. An application to a straight rod of circular cross is considered by means of a semidiscrete finite-element approximation that enables us to explore the effects of initial axial tension on the torsional, axial and flexural spectra and, furthermore, suggest paths toward estimation of the coefficients arising in the general theory.

Composite finite element approximation for nonlinear parabolic problems in nonconvex polygonal domains

We study a new class of finite elements so‐called composite finite elements (CFEs), introduced earlier by Hackbusch and Sauter, Numer. Math., 1997; 75:447‐472, for the approximation of nonlinear parabolic equation in a nonconvex polygonal domain. A two‐scale CFE discretization is used for the space discretizations, where the coarse‐scale grid discretized the domain at an appropriate distance from the boundary and the fine‐scale grid is used to resolve the boundary. A continuous, piecewise linear CFE space is employed for the spatially semidiscrete finite element approximation and the temporal discretizations is based on modified linearized backward Euler scheme. We derive almost optimal‐order convergence in space and optimal order in time for the CFE method in the L∞(L2) norm. Numerical experiment is carried out for an L‐shaped domain to illustrate our theoretical findings.

Semidiscrete finite element approximation of time optimal control problems for semilinear heat equations with nonsmooth initial data

This paper is devoted to the study of semidiscrete finite element approximation of time optimal control problem governed by semilinear heat equation with nonsmooth initial data. When the control is acted locally, an error estimate for the optimal time is obtained by making use of the approximation results for the equations. Moreover, with the help of Pontryagin’s maximum principle and the unique continuation property for the heat equation, a better error estimate for the optimal time can be derived when the control is acted globally into the state equation.

Gradient flow dynamics of two-phase biomembranes: Sharp interface variational formulation and finite element approximation

A finite element method for the evolution of a two-phase membrane in a sharp interface formulation is introduced. The evolution equations are given as an $L^2$--gradient flow of an energy involving an elastic bending energy and a line energy. In the two phases Helfrich-type evolution equations are prescribed, and on the interface, an evolving curve on an evolving surface, highly nonlinear boundary conditions have to hold. Here we consider both $C^0$-- and $C^1$--matching conditions for the surface at the interface. A new weak formulation is introduced, allowing for a stable semidiscrete parametric finite element approximation of the governing equations. In addition, we show existence and uniqueness for a fully discrete version of the scheme. Numerical simulations demonstrate that the approach can deal with a multitude of geometries. In particular, the paper shows the first computations based on a sharp interface description, which are not restricted to the axisymmetric case.

Error estimates of a semidiscrete finite element method for fractional stochastic diffusion‐wave equations

In this paper, we consider the Galerkin finite element method for solving the fractional stochastic diffusion‐wave equations driven by multiplicative noise, which can be used to describe the propagation of mechanical waves in viscoelastic media with random effects. The optimal strong convergence error estimates with respect to the semidiscrete finite element approximation in space are established. Finally, a numerical example is presented to verify the reliability of the theoretical study.

Fast evaluation and high accuracy finite element approximation for the time fractional subdiffusion equation

In this article, an efficient algorithm for the evaluation of the Caputo fractional derivative and the superconvergence property of fully discrete finite element approximation for the time fractional subdiffusion equation are considered. First, the space semidiscrete finite element approximation scheme for the constant coefficient problem is derived and supercloseness result is proved. The time discretization is based on the L1‐type formula, whereas the space discretization is done using, the fully discrete scheme is developed. Under some regularity assumptions, the superconvergence estimate is proposed and analyzed. Then, extension to the case of variable coefficients is also discussed. To reduce the computational cost, the fast evaluation scheme of the Caputo fractional derivative to solve the fractional diffusion equations is designed. Finally, numerical experiments are presented to support the theoretical results.

Finite element approximation for the dynamics of fluidic two-phase biomembranes

Biomembranes and vesicles consisting of multiple phases can attain a multitude of shapes, undergoing complex shape transitions. We study a Cahn--Hilliard model on an evolving hypersurface coupled to Navier--Stokes equations on the surface and in the surrounding medium to model these phenomena. The evolution is driven by a curvature energy, modelling the elasticity of the membrane, and by a Cahn--Hilliard type energy, modelling line energy effects. A stable semidiscrete finite element approximation is introduced and, with the help of a fully discrete method, several phenomena occurring for two-phase membranes are computed.

On the viscous Cahn-Hilliard-Navier-Stokes equations with dynamic boundary conditions

In the present article we study the viscous Cahn-Hilliard-Navier-Stokes model, endowed with dynamic boundary conditions, from the theoretical and numerical point of view. We start by deducing results on the existence, uniqueness and regularity of the solutions for the continuous problem. Then we propose a space semi-discrete finite element approximation of the model and we study the convergence of the approximate scheme. We also prove the stability and convergence of a fully discretized scheme, obtained using the semi-implicit Euler scheme applied to the space semi-discretization proposed previously. Numerical simulations are also presented to illustrate the theoretical results.

A discrete commutator theory for the consistency and phase error analysis of semi-discrete [formula omitted] finite element approximations to the linear transport equation

A novel method, based on a discrete commutator, for the analysis of consistency error and phase relations for semi-discrete continuous finite element approximation of the one-way wave equation is presented. The technique generalizes to arbitrary dimension, accommodates the use of compatible quadratures, does not require the use of complex calculations, is applicable on non-uniform mesh geometries, and is especially useful when conventional Taylor series or Fourier approaches are intractable. Following the theory the analysis method is demonstrated for several test cases.

Weak convergence for a spatial approximation of the nonlinear stochastic heat equation

We find the weak rate of convergence of the spatially semidiscrete finite element approximation of the nonlinear stochastic heat equation. Both multiplicative and additive noise is considered under different assumptions. This extends an earlier result of Debussche in which time discretization is considered for the stochastic heat equation perturbed by white noise. It is known that this equation has a solution only in one space dimension. In order to obtain results for higher dimensions, colored noise is considered here, besides white noise in one dimension. Integration by parts in the Malliavin sense is used in the proof. The rate of weak convergence is, as expected, essentially twice the rate of strong convergence.

Superconvergence for General Convex Optimal Control Problems Governed by Semilinear Parabolic Equations

We will investigate the superconvergence for the semidiscrete finite element approximation of distributed convex optimal control problems governed by semilinear parabolic equations. The state and costate are approximated by the piecewise linear functions and the control is approximated by piecewise constant functions. We present the superconvergence analysis for both the control variable and the state variables.