A finite element solution of diffraction problems in unbounded domains

In this paper, the method introduced in [3] is extended and applied to diffraction problems of acoustics and hydrodynamics. The problems dealt with are linear elliptic and may involve non-constant coefficients; they are set in unbounded domains. The method uses an integral representation formula with a regular kernel which allows an equivalent problem to be set in an interior annular closed region; it is shown how irregular frequencies are avoided. A variational formulation and its finite element discretization are detailed. Some numerical results are shown which support the validity of the technique and corroborate the theoretical analysis.     

Adaptive finite element heterogeneous multiscale method for homogenization problems

In this paper we present an a posteriori error analysis for elliptic homogenization problems discretized by the finite element heterogeneous multiscale method. Unlike standard finite element methods, our discretization scheme relies on macro- and microfinite elements. The desired macroscopic solution is obtained by a suitable averaging procedure based on microscopic data. As the macroscopic data (such as the macroscopic diffusion tensor) are not available beforehand, appropriate error indicators have to be defined for designing adaptive methods. We show that such indicators based only on the available macro- and microsolutions (used to compute the actual macrosolution) can be defined, allowing for a macroscopic mesh refinement strategy which is both reliable and efficient. The corresponding a posteriori estimates for the upper and lower bound are derived in the energy norm. In the case of a uniformly oscillating tensor, we recover the standard residual-based a posteriori error estimate for the finite element method applied to the homogenized problem. Numerical experiments confirm the efficiency and reliability of the adaptive multiscale method.     

On finite element approximation of general boundary value problems in nonlinear elasticity

This paper deals with the approximate solution of the general boundary value problem in nonlinear elasticity by the finite element displacement method. Under usual conditions which also guarantee the existence of locally unique solutions the quasi-optimal convergence inL ² andL ^∞ is shown for displacement fields and stresses. Furthermore a projective Newton method is considered which reduces the solution of the nonlinear continuous problem to the successive solution of a sequence of linearized problems of increasing dimension. It is proved that this procedure is well defined and also converges with quasi-optimal rates.     

Characteristic mixed finite element approximation of transient convection diffusion optimal control problems

In this paper, we investigate a characteristic mixed finite element approximation of transient convection diffusion optimal control problems. The state and the adjoint state are approximated by characteristic mixed finite element method, while the control is discretized by standard finite element method. We derive the continuous and discrete first-order optimality conditions and prove a priori error estimates for the state, the adjoint state and the control. Numerical examples are presented to illustrate the theoretical findings.     

A posteriori error estimates for space-time finite element approximation of quasistatic hereditary linear viscoelasticity problems

We give a space-time Galerkin finite element discretisation of the quasistatic compressible linear viscoelasticity problem as described by an elliptic partial differential equation with a fading memory Volterra integral. The numerical scheme consists of a continuous Galerkin approximation in space based on piecewise polynomials of degree p>0 (cG(p)), with a discontinuous Galerkin piecewise constant (dG(0)) or linear (dG(1)) approximation in time. A posteriori Galerkin-error estimates are derived by exploiting the Galerkin framework and optimal stability estimates for a related dual backward problem. The a posteriori error estimates are quite flexible: strong L_p-energy norms of the errors are estimated using time derivatives of the residual terms when the data are smooth, while weak-energy norms are used when the data are non-smooth (in time).We also give upper bounds on the dG(0)cG(1) a posteriori error estimates which indicate optimality. However, a complete analysis is not given.     

Application of a Galerkin finite element method to atmospheric transport problems

Numerical simulation of the movement of a contaminant within the atmosphere presents difficulties due to (a) The multi-dimensionality of the problem; (b) The fact that the horizontal transport is usually convection dominated; (c) The boundary conditions are mixed; (d) Both slow and fast atmospheric chemical reactions can be important. In this study, numerical experiments using a Crank-Nicolson Galerkin finite element method to solve the time-dependent partial differential equations demonstrate the applicability and accuracy of this method for the variety of conditions encountered in atmospheric pollutant modeling. The Crank-Nicolson Galerkin method using piecewise linear, piecewise cubic Hermite polynomials, and upwind finite elements is shown to accurately model the pure convection of initial wave forms. Numerical results studying the interactions of convection, diffusion, chemical reaction, pollutant removal, and the effects of contaminant emission source strength, source location and multiple sources are also presented.     

Spatial approximation of the radiation transport equation using a subgrid-scale finite element method

In this paper we present stabilized finite element methods to discretize in space the monochromatic radiation transport equation. These methods are based on the decomposition of the unknowns into resolvable and subgrid scales, with an approximation for the latter that yields a problem to be solved for the former. This approach allows us to design the algorithmic parameters on which the method depends, which we do here when the discrete ordinates method is used for the directional approximation. We concentrate on two stabilized methods, namely, the classical SUPG technique and the orthogonal subscale stabilization. A numerical analysis of the spatial approximation for both formulations is performed, which shows that they have a similar behavior: they are both stable and optimally convergent in the same mesh-dependent norm. A comparison with the behavior of the Galerkin method, for which a non-standard numerical analysis is done, is also presented.     

Algebraic flux correction for nonconforming finite element discretizations of scalar transport problems

This paper is concerned with the extension of the algebraic flux-correction (AFC) approach (Kuzmin in Computational fluid and solid mechanics, Elsevier, Amsterdam, pp 887–888, 2001; J Comput Phys 219:513–531, 2006; Comput Appl Math 218:79–87, 2008; J Comput Phys 228:2517–2534, 2009; Flux-corrected transport: principles, algorithms, and applications, 2nd edn. Springer, Berlin, pp 145–192, 2012; J Comput Appl Math 236:2317–2337, 2012; Kuzmin et al. in Comput Methods Appl Mech Eng 193:4915–4946, 2004; Int J Numer Methods Fluids 42:265–295, 2003; Kuzmin and Möller in Flux-corrected transport: principles, algorithms, and applications. Springer, Berlin, 2005; Kuzmin and Turek in J Comput Phys 175:525–558, 2002; J Comput Phys 198:131–158, 2004) to nonconforming finite element methods for the linear transport equation. Accurate nonoscillatory approximations to convection-dominated flows are obtained by stabilizing the continuous Galerkin method by solution-dependent artificial diffusion. Its magnitude is controlled by a flux limiter. This concept dates back to flux-corrected transport schemes. The unique feature of AFC is that all information is extracted from the system matrices which are manipulated to satisfy certain mathematical constraints. AFC schemes have been devised with conforming $P_1$ and $Q_1$ finite elements in mind but this is not a prerequisite. Here, we consider their extension to the nonconforming Crouzeix–Raviart element (Crouzeix and Raviart in RAIRO R3 7:33–76, 1973) on triangular meshes and its quadrilateral counterpart, the class of rotated bilinear Rannacher–Turek elements (Rannacher and Turek in Numer Methods PDEs 8:97–111, 1992). The underlying design principles of AFC schemes are shown to hold for (some variant of) both elements. However, numerical tests for a purely convective flow and a convection–diffusion problem demonstrate that flux-corrected solutions are overdiffusive for the Crouzeix–Raviart element. Good resolution of smooth and discontinuous profiles is attested to $Q_1^\mathrm{nc}$ approximations on quadrilateral meshes. A synthetic benchmark is used to quantify the artificial diffusion present in conforming and nonconforming high-resolution schemes of AFC-type. Finally, the implementation of efficient sparse matrix–vector multiplications is addressed.     

The finite element approximation of Hamilton-Jacobi-Bellman equations

This paper deals with the finite element approximation of Hamilton-Jacobi-Bellman equations. We establish a convergence and a quasi-optimal L^∞-error estimate, involving a weakly coupled systems of quasi-variational inequalities for the solution of which an interative scheme of monotone kind is introduced and analyzed.     

WEB-Spline-based mesh-free finite element approximation for p-Laplacian

In this paper, we discuss the approximation of p-Laplace problem using WEB-Spline based mesh free finite elements. Along with usual weak formulation, we also consider the mixed formulation of the p-Laplace problem. We give existence, uniqueness results for both continuous and discrete problems. We also provide a priori error estimates for both the formulations.     

A parallel finite element procedure for contact-impact problems

An efficient parallel finite element procedure for contact-impact problems is presented within the framework of explicit finite element analysis with thepenalty method. The procedure concerned includes a parallel Belytschko-Lin-Tsay shell element generation algorithm and a parallel contact-impact algorithm based on the master-slave slideline algorithm. An element-wise domain decomposition strategy and a communication minimization strategy are featured to achieve almost perfect load balancing among processors and to show scalability of the parallel performance. Throughout this work, a prototype code, named GT-PARADYN, is developed on the IBM SP2 to implement the procedure presented, under message-passing paradigm. Some examples are provided to demonstrate the timing results of the algorithms, discussing the accuracy and efficiency of the code.     

A finite element approximation of a variational inequality related to hydraulics

In this paper we study the convergence of a finite element approximation for a variational inequality related to hydraulics and we prove, for both linear and quadratic elements, error bounds in terms of the mesh size and a theorem of convergence of the domains This work is based in part on the graduation thesis of G. Sacchi discussed in october 1974 at University of Pavia.     

Generalized finite element method using mesh-based handbooks: application to problems in domains with many voids

This paper describes a new version of the generalized finite element method, originally developed [Int. J. Numer. Methods Engrg. 47 (2000) 1401; Comput. Methods Appl. Mech. Engrg. 181 (2000) 43; The design and implementation of the generalized finite element method, Ph.D. thesis, Texas A&M University, College Station, Texas, August 2000; Comput. Methods Appl. Mech. Engrg. 190 (2001) 4081], which is well suited for problems set in domains with a large number of internal features (e.g. voids, inclusions, cracks, etc.). The main idea is to employ handbook functions constructed on subdomains resulting from the mesh-discretization of the problem domain. The proposed new version of the GFEM is shown to be robust with respect to the spacing of the features and is capable of achieving high accuracy on meshes which are rather coarse relative to the distribution of the features.     

A viscosity approximation scheme for finite mixed equilibrium problems and variational inequality problems and fixed point problems

In this paper, we introduce a new iterative scheme for finding a common element of the set of solutions of finite mixed equilibrium problems, the set of solutions of variational inequalities for two cocoercive mappings, the set of common fixed points of an infinite family of nonexpansive mappings and the set of common fixed points of a nonexpansive semigroup in Hilbert space. Then we prove a strong convergence theorem under some suitable conditions. The results obtained in this paper extend and improve many recent ones announced by many others.     

A high-continuity finite element model for two-dimensional elastic problems

A finite element model for the analysis of two-dimensional elastic problems is presented. The proposed discretization is based on a biquadratic interpolation for the displacement components and takes advantage of the enforcement of the interelement continuity to obtain a profitable reduction of the total number of the degrees of freedom. One node (two kinematical parameters) per element only is required.Numerical results obtained for some test problems show the accuracy of the model in analyzing both the deformations and the stress distribution.     

A quasioptimal finite element method for elliptic interface problems

In [1] Babu?ka proposed perturbed variational methods for elliptic problems, with discontinuous coefficients. However, these methods are not quasioptimal, i.e. the approximate solutions generated by such methods do not reproduce the properties of “best approximation” possessed by the subspace of admissable approximants. In this paper we consider certain extrapolates obtained by use of a particular method of [1] and obtain “optimal” asymptotic error estimates. Our approach is similar to that of [7] where we proposed extrapolation methods for elliptic problems with smooth coefficients.     

Mimetic finite difference approximation of quasilinear elliptic problems

In this work we approximate the solution of a quasilinear elliptic problem of monotone type by using the Mimetic Finite Difference (MFD) method. Under a suitable approximation assumption, we prove that the MFD approximate solution converges, with optimal rate, to the exact solution in a mesh-dependent energy norm. The resulting nonlinear discrete problem is then solved iteratively via linearization by applying the Ka?anov method. The convergence of the Ka?anov algorithm in the discrete mimetic framework is also proved. Several numerical experiments confirm the theoretical analysis.