Finite element approximation of -surfaces

In this paper a piecewise linear finite element approximation of -surfaces, or surfaces with constant mean curvature, spanned by a given Jordan curve in is considered. It is proved that the finite element -surfaces converge to the exact -surfaces under the condition that the Jordan curve is rectifiable. Several numerical examples are given.

     

Finite element approximation for equations of magnetohydrodynamics

We consider the equations of stationary incompressible magnetohydrodynamics posed in three dimensions, and treat the full coupled system of equations with inhomogeneous boundary conditions. We prove the existence of solutions without any conditions on the data. Also we discuss a finite element discretization and prove the existence of a discrete solution, again without any conditions on the data. Finally, we derive error estimates for the nonlinear case.

     

Finite element approximation of a microstructure evolution

The evolution model of a microstructure described as appropriately generalized Young measures, which was developed in [11], is discretized here by means of a suitably adapted finite element method. The convergence of the approximate solutions is proved, and a one-dimensional example is treated to discuss some implementation experience and to show some illustrative results.     

Finite element approximation of nonlocal parabolic problem

In this article, we present a finite element scheme combined with backward Euler method to solve a nonlocal parabolic problem. An important issue in the numerical solution of nonlocal problems while using Newton's method is related to its structure. In fact differently from the local case where the Jacobian matrix is sparse and banded, in the nonlocal case the Jacobian matrix is dense and computations are much more onerous compared to that for differential equations. In order to avoid this difficulty, we use the technique given by Gudi (SIAM J Numer Anal 50 (2012), 657–668) for elliptic nonlocal problem of Kirchhoff type. We discuss the well‐posedness of the weak formulation at continuous as well as at discrete levels. We also derive a priori error estimates for semidiscrete and fully discrete formulations in L² and H¹ norms. Results based on the usual finite element method are provided to confirm the theoretical estimates. © 2016 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 33: 786–813, 2017     

Finite element approximation of the volume-matching problem

Summary Optimal orderH ¹ andL error bounds are obtained for a continuous piecewise linear finite element approximation of the volume matching problem. This problem consists of minimising |v| _1, ² overvH ¹() subject to the inequality constraintv0 and a number of linear equality constraints. The presence of the equality constraints leads to Lagrange multipliers, which in turn lead to complications with the standard error analysis for variational inequalities. Finally we consider an algorithm for solving the resulting algebraic problem.Supported by a SERC research studentship     

Finite element approximation of the Sobolev constant

Denoting by S the sharp constant in the Sobolev inequality in W₀^1,2(B){{\rm W}_0^{1,2}(B)}, being B the unit ball in \mathbbR³{\mathbb{R}^3}, and denoting by S _h its approximation in a suitable finite element space, we show that S _h converges to S as h\searrow0{h\searrow0} with a polynomial rate of convergence. We provide both an upper and a lower bound on the rate of convergence, and present some numerical results.     

Finite element approximation of large bending isometries

A finite element scheme for the approximation of large isometric deformations with minimal bending energy is devised and analyzed. The convergence to a stationary point and energy decreasing property of an iterative algorithm for the numerical solution of the scheme is proved. Numerical experiments illustrate the performance of the iteration and show that the discretization leads to accurate approximations for large vertical loads and compressive boundary conditions.     

Finite element approximation of viscoelastic progressively incompressible flows

Summary To avoid any numerical locking in the finite element approximation of viscoelastic flow problems, we propose a three-field approximation of this problem. This approximation, which involves velocities, stresses, and pressures is proved to converge for all times. In the proof, we also obtain convergence results for the three-fields finite element approximation of incompressible elasticity problems.     

Finite element approximation of a prestressed shell model

This work deals with the finite element approximation of a prestressed shell model formulated in Cartesian coordinates system. The considered constrained variational problem is not necessarily positive. Moreover, because of the constraint, it cannot be discretized by conforming finite element methods. A penalized version of the model and its discretization are then proposed. We prove existence and uniqueness results of solutions for the continuous and discrete problems, and we derive optimal a priori error estimates. Numerical tests that validate and illustrate our approach are given.     

Finite element approximation to nonlinear coupled thermal problem

A nonlinear coupled elliptic system modelling a large class of engineering problems was discussed in [A.F.D. Loula, J. Zhu, Finite element analysis of a coupled nonlinear system, Comp. Appl. Math. 20 (3) (2001) 321–339; J. Zhu, A.F.D. Loula, Mixed finite element analysis of a thermally nonlinear coupled problem, Numer. Methods Partial Differential Equations 22 (1) (2006) 180–196]. The convergence analysis of iterative finite element approximation to the solution was done under an assumption of ‘small’ solution or source data which guarantees the uniqueness of the nonlinear coupled system. Generally, a nonlinear system may have multiple solutions. In this work, the regularity of the weak solutions is further studied. The nonlinear finite element approximations to the nonsingular solutions are then proposed and analyzed. Finally, the optimal order error estimates in H¹$H^1$-norm and L²$L^2$-norm as well as in W^1,p$W^{1,p}$-norm and L^p$L^p$-norm are obtained.     

Finite element approximation for time-dependent diffusion with measure-valued source

The convergence of finite element methods for elliptic and parabolic partial differential equations is well-established if source terms are sufficiently smooth. Noting that finite element computation is easily implemented even when the source terms are measure-valued—for instance, modeling point sources by Dirac delta distributions—we prove new convergence order results in two and three dimensions both for elliptic and for parabolic equations with measures as source terms. These analytical results are confirmed by numerical tests using COMSOL Multiphysics.     

Finite element approximation for a modified anomalous subdiffusion equation

In this paper, we consider a modified anomalous subdiffusion equation (MASFE) for describing processes that become less anomalous as time progresses by the inclusion of a second fractional time derivative acting on the diffusion term. Firstly, a semi-discrete approximation for the MASFE is proposed. The stability and convergence of the semi-discrete approximation are discussed. Secondly, a finite element approximation for the MASFE is derived. The stability and convergence of the finite element approximation are investigated, respectively. Finally, some numerical examples are presented to demonstrate the effectiveness of theoretical analysis.     

Finite element approximation of diffusion equations with convolution terms

Approximation of solutions to diffusion equations with memory represented by convolution integral terms is considered. Such problems arise from modeling of flows in fissured media. Convergence of the method is proved and results of numerical experiments confirming the theoretical results are presented. The advantages of implementation of the algorithm in a multiprocessing environment are discussed.

     

Finite element network approximation of conductivity in particle composites

A new finite element method computes conductivity in some unstructured particle-reinforced composite material. The 2-phase material under consideration is composed of a poorly conducting matrix material filled by highly conducting circular inclusions which are randomly dispersed. The mathematical model is a Poisson-type problem with discontinuous coefficients. The discontinuities are huge in contrast and quantity. The proposed method generalizes classical continuous piecewise affine finite elements to special computational meshes which encode the particles in a network structure. Important geometric parameters such as the volume fraction are preserved exactly. The computational complexity of the method is (almost) proportional to the number of inclusions. This is minimal in the sense that the representation of the underlying geometry via the positions and radii of the inclusions is of the same complexity. The discretization error is proportional to the distance of neighboring inclusions and independent of the conductivity contrast in the medium.     

Finite element approximation of an Allen-Cahn/Cahn-Hilliard system

We consider an Allen–Cahn/Cahn–Hilliard system witha non-degenerate mobility and (i) a logarithmic free energyand (ii) a non-smooth free energy (the deep quench limit). Thissystem arises in the modelling of phase separation and orderingin binary alloys. In particular we prove in each case that thereexists a unique solution for sufficiently smooth initial data.Further, we prove an error bound for a fully practical piecewiselinear finite element approximation of (i) and (ii) in one andtwo space dimensions (and three space dimensions for constantmobility). The error bound being optimal in the deep quenchlimit. In addition an iterative scheme for solving the resultingnonlinear discrete system is analysed. Finally some numericalexperiments are presented.     

Finite element approximation of a free boundary plasma problem

In this article, we study a finite element approximation for a model free boundary plasma problem. Using a mixed approach (which resembles an optimal control problem with control constraints), we formulate a weak formulation and study the existence and uniqueness of a solution to the continuous model problem. Using the same setting, we formulate and analyze the discrete problem. We derive optimal order energy norm a priori error estimates proving the convergence of the method. Further, we derive a reliable and efficient a posteriori error estimator for the adaptive mesh refinement algorithm. Finally, we illustrate the theoretical results by some numerical examples.     

Finite element approximation of a nonlinear cross-diffusion population model

Summary. We consider a fully discrete finite element approximation of the nonlinear cross-diffusion population model: Find u_i, the population of the i^th species, i=1 and 2, such that where ji and g_i(u₁,u₂):=(_i–_iiu_i–_iju_j)u_i. In the above, the given data is as follows: v is an environmental potential, c_i, a_i are diffusion coefficients, b_i are transport coefficients, _i are the intrinsic growth rates, and _ii are intra-specific, whereas _ij, ij, are interspecific competition coefficients. In addition to showing well-posedness of our approximation, we prove convergence in space dimensions d3. Finally some numerical experiments in one space dimension are presented.Mathematics Subject Classification (2000): 65M60, 65M12, 35K55, 92D25Acknowledgements. Part of this work was carried out while the authors participated in the 2003 programme {\it Computational Challenges in Partial Differential Equations} at the Isaac Newton Institute, Cambridge, UK.     

Finite element approximation with numerical integration for differential eigenvalue problems

Error estimates of the finite element method with numerical integration for differential eigenvalue problems are presented. More specifically, refined results on the eigenvalue dependence for the eigenvalue and eigenfunction errors are proved. The theoretical results are illustrated by numerical experiments for a model problem.     

Finite element approximation of spectral acoustic problems on curved domains

Summary This paper deals with the finite element approximation of the displacement formulation of the spectral acoustic problem on a curved non convex two-dimensional domain . Convergence and error estimates are proved for Raviart-Thomas elements on a discrete polygonal domain _h in the framework of the abstract spectral approximation theory. Similar results have been previously proved only for polygonal domains. Numerical tests confirming the theoretical results are reported.Mathematics Subject Classification (2000):65N25, 65N30, 70J30Supported by FONDECYT 2000114 (Chile)Supported by FONDAP in Applied Mathematics (Chile)