A fully adaptive numerical approximation for a two-dimensional epidemic model with nonlinear cross-diffusion

An epidemic model is formulated by a reaction–diffusion system where the spatial pattern formation is driven by cross-diffusion. The reaction terms describe the local dynamics of susceptible and infected species, whereas the diffusion terms account for the spatial distribution dynamics. For both self-diffusion and cross-diffusion, nonlinear constitutive assumptions are suggested. To simulate the pattern formation two finite volume formulations are proposed, which employ a conservative and a non-conservative discretization, respectively. An efficient simulation is obtained by a fully adaptive multiresolution strategy. Numerical examples illustrate the impact of the cross-diffusion on the pattern formation.     

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 of a sixth order nonlinear degenerate parabolic equation

Summary. We consider a finite element approximation of the sixth order nonlinear degenerate parabolic equation where generically for any given In addition to showing well-posedness of our approximation, we prove convergence in space dimensions $d \leq 3$. Furthermore an iterative scheme for solving the resulting nonlinear discrete system is analysed. Finally some numerical experiments in one and two space dimensions are presented. Mathematics Subject Classification (2000): 65M60, 65M12, 35K55, 35K65, 35K35Supported by the EPSRC, U.K. through grant GR/M29689.Supported by the EPSRC, and by the DAAD through a Doktorandenstipendium     

Finite element approximation of a fourth order nonlinear degenerate parabolic equation

Summary. We consider a fully practical finite element approximation of the fourth order nonlinear degenerate parabolic equation where generically for any given . An iterative scheme for solving the resulting nonlinear discrete system is analysed. In addition to showing well-posedness of our approximation, we prove convergence in one space dimension. Finally some numerical experiments are presented. Received July 29, 1997     

Finite element approximation of a problem with a nonlinear Newton boundary condition

The paper is concerned with the study of an elliptic boundary value problem with a nonlinear Newton boundary condition. The existence and uniqueness of the solution of the continuous problem is established with the aid of the monotone operator theory. The main attention is paid to the investigation of the finite element approximation using numerical integration for the computation of nonlinear boundary integrals. The solvability of the discrete finite element problem is proved and the convergence of the approximate solutions to the exact one is analysed. Received April 15, 1996 / Revised version received November 22, 1996     

Finite element approximation to a contact problem for a nonlinear thermoviscoelastic beam

In this paper we propose and analyze a finite element method to the solution of a quasi-static contact problem between a nonlinear beam and a rigid obstacle. Error estimates and energy decay are obtained and some numerical simulations described.     

Semi-discretization in time and numerical convergence of solutions of a nonlinear cross-diffusion population model

Summary. A positivity-preserving numerical scheme for a strongly coupled cross-diffusion model for two competing species is presented, based on a semi-discretization in time. The variables are the population densities of the species. Existence of strictly positive weak solutions to the semidiscrete problem is proved. Moreover, it is shown that the semidiscrete solutions converge to a non-negative solution of the continuous system in one space dimension. The proofs are based on a symmetrization of the problem via an exponential transformation of variables and the use of an entropy functional. Received September 10, 2001 / Revised version received February 25, 2002 / Published online June 17, 2002     

Finite element approximation

We study a discretization procedure, using finite elements, for a class of non-isotropic free-discontinuity problems based on the non-local approximation proposed in [18]     

Finite element approximation of a periodic Ginzburg-Landau model for type-II superconductors

Summary We consider efficient finite element algorithms for the computational simulation of type-II superconductors. The algorithms are based on discretizations of a periodic Ginzburg-Landau model. Periodicity is defined with respect to a non-orthogonal lattice that is not necessarily aligned with the coordinate axes; also, the primary dependent variables employed in the model satisfy non-standard quasi-periodic boundary conditions. After introducing the model, we define finite element schemes, derive error estimates of optimal order, and present the results of some numerical calculations. For a similar quality of simulation, the resulting algorithms seem to be significantly less costly than are previously used numerical approximation methods.     

Finite element approximations of a nonlinear diffusion model with memory

The convergence of a finite element scheme approximating a nonlinear system of integro-differential equations is proven. This system arises in mathematical modeling of the process of a magnetic field penetrating into a substance. Properties of existence, uniqueness and asymptotic behavior of the solutions are briefly described. The decay of the numerical solution is compared with both the theoretical and finite difference results.     

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.     

Analysis of a parabolic cross-diffusion population model without self-diffusion

The global existence of non-negative weak solutions to a strongly coupled parabolic system arising in population dynamics is shown. The cross-diffusion terms are allowed to be arbitrarily large, whereas the self-diffusion terms are assumed to disappear. The last assumption complicates the analysis since these terms usually provide H¹ estimates of the solutions. The existence proof is based on a positivity-preserving backward Euler-Galerkin approximation, discrete entropy estimates, and L¹ weak compactness arguments. Furthermore, employing the entropy-entropy production method, we show for special stationary solutions that the transient solution converges exponentially fast to its steady state. As a by-product, we prove that only constant steady states exist if the inter-specific competition parameters disappear no matter how strong the cross-diffusion constants are.     

Finite element approximation of a phase field model arising in nanostructure patterning

We consider a fully practical finite element approximation of the nonlinear parabolic Cahn–Hilliard system subject to an initial condition on the conserved order parameter , and mixed boundary conditions. Here, is the interfacial parameter, is the field strength parameter, is the obstacle potential, is the diffusion coefficient, and denotes differentiation with respect to the second argument. Furthermore, w is the chemical potential and is the electrostatic potential. The system, in the context of nanostructure patterning, has been proposed to model the manipulation of morphologies in organic solar cells with the help of an applied electric field. In the limit , it reduces to a sharp interface problem that models the evolution of an unstable interface between two dielectric media in the presence of a quasistatic electric field. On introducing a finite element approximation for the above Cahn–Hilliard system, we prove existence and stability of a discrete solution. Moreover, in the case of two space dimensions, we are able to prove convergence and hence existence of a solution to the considered system of partial differential equations. We demonstrate the practicality of our finite element approximation with several numerical simulations in two and three space dimensions. © 2015 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 31: 1890–1924, 2015     

Finite difference approximation of a nonlinear integro-differential system

Finite difference approximation of the nonlinear integro-differential system associated with the penetration of a magnetic field into a substance is studied. The convergence of the finite difference scheme is proved. The rate of convergence of the discrete scheme is given. The decay of the numerical solution is compared with the analytical results proven earlier.     

Finite element analysis for a nonlinear diffusion model in image processing

The optimal error estimate O(h^k+1) for a popular nonlinear diffusion model widely used in image processing is proved for the standard k^th-order (k ≥ 1) conforming tensor-product finite elements in the L²-norm. The optimal L²-estimate is obtained by the integral identity technique [1–3] without using the classic Nitsche duality argument [4].     

Finite dimensional approximation of nonlinear problems

Summary In the first two papers of this series [4, 5], we have studied a general method of approximation of nonsingular solutions and simple limit points of nonlinear equations in a Banach space. We derive here general approximation results of the branches of solutions in the neighborhood of a simple bifurcation point. The abstract theory is applied to the Galerkin approximation of nonlinear variational problems and to a mixed finite element approximation of the von Kármán equations.The work of F. Brezzi has been completed during his stay at the Université P. et M. Curie and at the Ecole PolytechniqueThe work of J. Rappaz has been supported by the Fonds National Suisse de la Recherche Scientifique