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## Space Dimensions (space + dimension)
Kinds of Space Dimensions
## Selected Abstracts## A space,time discontinuous Galerkin method for the solution of the wave equation in the time domain INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, Issue 3 2009Steffen PetersenAbstract In recent years, the focus of research in the field of computational acoustics has shifted to the medium frequency regime and multiscale wave propagation. This has led to the development of new concepts including the discontinuous enrichment method. Its basic principle is the incorporation of features of the governing partial differential equation in the approximation. In this contribution, this concept is adapted for the simulation of transient problems governed by the wave equation. We present a space,time discontinuous Galerkin method with Lagrange multipliers, where the shape approximation in space and time is based on solutions of the homogeneous wave equation. The use of hierarchical wave-like basis functions is enabled by means of a variational formulation that allows for discontinuities in both the spatial and the temporal discretizations. Numerical examples in one space dimension demonstrate the outstanding performance of the proposed method compared with conventional space,time finite element methods. Copyright © 2008 John Wiley & Sons, Ltd. [source] ## Some recent finite volume schemes to compute Euler equations using real gas EOS INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDS, Issue 12 2002T. GallouëtAbstract This paper deals with the resolution by finite volume methods of Euler equations in one space dimension, with real gas state laws (namely, perfect gas EOS, Tammann EOS and Van Der Waals EOS). All tests are of unsteady shock tube type, in order to examine a wide class of solutions, involving Sod shock tube, stationary shock wave, simple contact discontinuity, occurrence of vacuum by double rarefaction wave, propagation of a one-rarefaction wave over ,vacuum', , Most of the methods computed herein are approximate Godunov solvers: VFRoe, VFFC, VFRoe ncv (,, u, p) and PVRS. The energy relaxation method with VFRoe ncv (,, u, p) and Rusanov scheme have been investigated too. Qualitative results are presented or commented for all test cases and numerical rates of convergence on some test cases have been measured for first- and second-order (Runge,Kutta 2 with MUSCL reconstruction) approximations. Note that rates are measured on solutions involving discontinuities, in order to estimate the loss of accuracy due to these discontinuities. Copyright © 2002 John Wiley & Sons, Ltd. [source] ## One- and Two-Component Bottle-Brush Polymers: Simulations Compared to Theoretical Predictions MACROMOLECULAR THEORY AND SIMULATIONS, Issue 7 2007Hsiao-Ping HsuAbstract Scaling predictions for bottle-brush polymers with a rigid backbone and flexible side chains under good solvent conditions are discussed and their validity is assessed by a comparison with Monte Carlo simulations of a simple lattice model. It is shown that typically only a rather weak stretching of the side chains is realized, and then the scaling predictions are not applicable. Also two-component bottle brush polymers are considered, where two types (A,B) of side chains are grafted, assuming that monomers of different kind repel each other. In this case, variable solvent quality is allowed. Theories predict "Janus cylinder"-type phase separation along the backbone in this case. The Monte Carlo simulations, using the pruned-enriched Rosenbluth method (PERM) give evidence that the phase separation between an A-rich part of the cylindrical molecule and a B-rich part can only occur locally. The correlation length of this microphase separation can be controlled by the solvent quality. This lack of a phase transition is interpreted by an analogy with models for ferromagnets in one space dimension. [source] ## The transient equations of viscous quantum hydrodynamics MATHEMATICAL METHODS IN THE APPLIED SCIENCES, Issue 4 2008Michael DreherAbstract We study the viscous model of quantum hydrodynamics in a bounded domain of space dimension 1, 2, or 3, and in the full one-dimensional space. This model is a mixed-order partial differential system with nonlocal and nonlinear terms for the particle density, current density, and electric potential. By a viscous regularization approach, we show existence and uniqueness of local in time solutions. We propose a reformulation as an equation of Schrödinger type, and we prove the inviscid limit. Copyright © 2007 John Wiley & Sons, Ltd. [source] ## Asymmetric invariants for a class of strictly hyperbolic systems including the Timoshenko beam MATHEMATICAL METHODS IN THE APPLIED SCIENCES, Issue 11 2007Clelia MarchionnaAbstract We introduce a set of conserved quantities of energy-type for a strictly hyperbolic system of two coupled wave equations in one space dimension. The system is subject to mechanical boundary conditions. Some of these invariants are asymmetric in the sense that their defining quadratic form contains second order derivatives in only one of the unknowns. We study their independence with respect to the usual energies and characterize their sign. In many cases, our results provide sharp well-posedness and stability results. Finally, we apply some of our conservation laws to the study of a singular perturbation problem previously considered by J. Lagnese and J. L. Lions. Copyright © 2007 John Wiley & Sons, Ltd. [source] ## On non-Newtonian incompressible fluids with phase transitions MATHEMATICAL METHODS IN THE APPLIED SCIENCES, Issue 13 2006Namkwon KimAbstract A modified model for a binary fluid is analysed mathematically. The governing equations of the motion consists of a Cahn,Hilliard equation coupled with a system describing a class of non-Newtonian incompressible fluid with p -structure. The existence of weak solutions for the evolution problems is shown for the space dimension d=2 with p, 2 and for d=3 with p, 11/5. The existence of measure-valued solutions is obtained for d=3 in the case 2, p< 11/5. Similar existence results are obtained for the case of nondifferentiable free energy, corresponding to the density constraint |,| , 1. We also give regularity and uniqueness results for the solutions and characterize stable stationary solutions. Copyright © 2006 John Wiley & Sons, Ltd. [source] ## Thermoelasticity with second sound,exponential stability in linear and non-linear 1-d MATHEMATICAL METHODS IN THE APPLIED SCIENCES, Issue 5 2002Reinhard RackeWe consider linear and non-linear thermoelastic systems in one space dimension where thermal disturbances are modelled propagating as wave-like pulses travelling at finite speed. This removal of the physical paradox of infinite propagation speed in the classical theory of thermoelasticity within Fourier's law is achieved using Cattaneo's law for heat conduction. For different boundary conditions, in particular for those arising in pulsed laser heating of solids, the exponential stability of the now purely, but slightly damped, hyperbolic linear system is proved. A comparison with classical hyperbolic,parabolic thermoelasticity is given. For Dirichlet type boundary conditions,rigidly clamped, constant temperature,the global existence of small, smooth solutions and the exponential stability are proved for a non-linear system. Copyright © 2002 John Wiley & Sons, Ltd. [source] ## Behavior of the solution of a random semilinear heat equation COMMUNICATIONS ON PURE & APPLIED MATHEMATICS, Issue 9 2008S. R. S. VaradhanWe consider a semilinear heat equation in one space dimension, with a random source at the origin. We study the solution, which describes the equilibrium of this system, and prove that, as the space variable tends to infinity, the solution becomes a.s. asymptotic to a steady state. We also study the fluctuations of the solution around the steady state. [source] ## A quantitative compactness estimate for scalar conservation laws COMMUNICATIONS ON PURE & APPLIED MATHEMATICS, Issue 7 2005Camillo de LellisIn the case of a scalar conservation law with convex flux in space dimension one, P. D. Lax proved [Comm. Pure and Appl. Math.7 (1954)] that the semigroup defining the entropy solution is compact in L for each positive time. The present note gives an estimate of the ,-entropy in L of the set of entropy solutions at time t > 0 whose initial data run through a bounded set in L1. © 2005 Wiley Periodicals, Inc. [source] ## Low-curvature image simplifiers: Global regularity of smooth solutions and Laplacian limiting schemes COMMUNICATIONS ON PURE & APPLIED MATHEMATICS, Issue 6 2004Andrea L. BertozziWe consider a class of fourth-order nonlinear diffusion equations motivated by Tumblin and Turk's "low-curvature image simplifiers" for image denoising and segmentation. The PDE for the image intensity u is of the form where g(s) = k2/(k2 + s2) is a "curvature" threshold and , denotes a fidelity-matching parameter. We derive a priori bounds for ,u that allow us to prove global regularity of smooth solutions in one space dimension, and a geometric constraint for finite-time singularities from smooth initial data in two space dimensions. This is in sharp contrast to the second-order Perona-Malik equation (an ill-posed problem), on which the original LCIS method is modeled. The estimates also allow us to design a finite difference scheme that satisfies discrete versions of the estimates, in particular, a priori bounds on the smoothness estimator in both one and two space dimensions. We present computational results that show the effectiveness of such algorithms. Our results are connected to recent results for fourth-order lubrication-type equations and the design of positivity-preserving schemes for such equations. This connection also has relevance for other related fourth-order imaging equations. © 2004 Wiley Periodicals, Inc. [source] ## Linear vs. nonlinear selection for the propagation speed of the solutions of scalar reaction-diffusion equations invading an unstable equilibrium COMMUNICATIONS ON PURE & APPLIED MATHEMATICS, Issue 5 2004Marcello LuciaWe revisit the classical problem of speed selection for the propagation of disturbances in scalar reaction-diffusion equations with one linearly stable and one linearly unstable equilibrium. For a wide class of initial data this problem reduces to finding the minimal speed of the monotone traveling wave solutions connecting these two equilibria in one space dimension. We introduce a variational characterization of these traveling wave solutions and give a necessary and sufficient condition for linear versus nonlinear selection mechanism. We obtain sufficient conditions for the linear and nonlinear selection mechanisms that are easily verifiable. Our method also allows us to obtain efficient lower and upper bounds for the propagation speed. © 2004 Wiley Periodicals, Inc. [source] ## Semiclassical limit for the Schrödinger-Poisson equation in a crystal COMMUNICATIONS ON PURE & APPLIED MATHEMATICS, Issue 7 2001Philippe BechoucheWe give a mathematically rigorous theory for the limit from a weakly nonlinear Schrödinger equation with both periodic and nonperiodic potential to the semiclassical version of the Vlasov equation. To this end we perform simultaneously a classical limit (vanishing Planck constant) and a homogenization limit of the periodic structure (vanishing lattice length taken proportional to the Planck constant). We introduce a new variant of Wigner transforms, namely the "Wigner Bloch series" as an adaption of the Wigner series for density matrices related to two different "energy bands." Another essential tool are estimates on the commutators of the projectors into the Floquet subspaces ("band subspaces") and the multiplicative potential operator that destroy the invariance of these band subspaces under the periodic Hamiltonian. We assume the initial data to be concentrated in isolated bands but allow for band crossing of the other bands which is the generic situation in more than one space dimension. The nonperiodic potential is obtained from a coupling to the Poisson equation, i.e., we take into account the self-consistent Coulomb interaction. Our results hold also for the easier linear case where this potential is given. We hence give the first rigorous derivation of the (nonlinear) "semiclassical equations" of solid state physics widely used to describe the dynamics of electrons in semiconductors. © 2001 John Wiley & Sons, Inc. [source] ## Conjugate filter approach for shock capturing, INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN BIOMEDICAL ENGINEERING, Issue 2 2003Yun GuAbstract This paper introduces a new scheme for the numerical computation involving shock waves. The essence of the scheme is to adaptively implement a conjugate low-pass filter to effectively remove the accumulated numerical errors produced by a set of high-pass filters. The advantages of using such an adaptive algorithm are its controllable accuracy, relatively low cost and easy implementation. Numerical examples in one and two space dimensions are presented to illustrate the proposed scheme. Copyright © 2003 John Wiley & Sons, Ltd. [source] ## An adaptive multiresolution method for parabolic PDEs with time-step control INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, Issue 6 2009M. O. DominguesAbstract We present an efficient adaptive numerical scheme for parabolic partial differential equations based on a finite volume (FV) discretization with explicit time discretization using embedded Runge,Kutta (RK) schemes. A multiresolution strategy allows local grid refinement while controlling the approximation error in space. The costly fluxes are evaluated on the adaptive grid only. Compact RK methods of second and third order are then used to choose automatically the new time step while controlling the approximation error in time. Non-admissible choices of the time step are avoided by limiting its variation. The implementation of the multiresolution representation uses a dynamic tree data structure, which allows memory compression and CPU time reduction. This new numerical scheme is validated using different classical test problems in one, two and three space dimensions. The gain in memory and CPU time with respect to the FV scheme on a regular grid is reported, which demonstrates the efficiency of the new method. Copyright © 2008 John Wiley & Sons, Ltd. [source] ## Adaptive finite elements with large aspect ratio for mass transport in electroosmosis and pressure-driven microflows INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDS, Issue 9 2010Virabouth PrachitthamAbstract A space,time adaptive method is presented for the numerical simulation of mass transport in electroosmotic and pressure-driven microflows in two space dimensions. The method uses finite elements with large aspect ratio, which allows the electroosmotic flow and the mass transport to be solved accurately despite the presence of strong boundary layers. The unknowns are the external electric potential, the electrical double layer potential, the velocity field and the sample concentration. Continuous piecewise linear stabilized finite elements with large aspect ratio and the Crank,Nicolson scheme are used for the space and time discretization of the concentration equation. Numerical results are presented showing the efficiency of this approach, first in a straight channel, then in crossing and multiple T-form configuration channels. Copyright © 2009 John Wiley & Sons, Ltd. [source] ## Numerical simulation of bubble and droplet deformation by a level set approach with surface tension in three dimensions INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDS, Issue 9 2010Roberto CroceAbstract In this paper we present a three-dimensional Navier,Stokes solver for incompressible two-phase flow problems with surface tension and apply the proposed scheme to the simulation of bubble and droplet deformation. One of the main concerns of this study is the impact of surface tension and its discretization on the overall convergence behavior and conservation properties. Our approach employs a standard finite difference/finite volume discretization on uniform Cartesian staggered grids and uses Chorin's projection approach. The free surface between the two fluid phases is tracked with a level set (LS) technique. Here, the interface conditions are implicitly incorporated into the momentum equations by the continuum surface force method. Surface tension is evaluated using a smoothed delta function and a third-order interpolation. The problem of mass conservation for the two phases is treated by a reinitialization of the LS function employing a regularized signum function and a global fixed point iteration. All convective terms are discretized by a WENO scheme of fifth order. Altogether, our approach exhibits a second-order convergence away from the free surface. The discretization of surface tension requires a smoothing scheme near the free surface, which leads to a first-order convergence in the smoothing region. We discuss the details of the proposed numerical scheme and present the results of several numerical experiments concerning mass conservation, convergence of curvature, and the application of our solver to the simulation of two rising bubble problems, one with small and one with large jumps in material parameters, and the simulation of a droplet deformation due to a shear flow in three space dimensions. Furthermore, we compare our three-dimensional results with those of quasi-two-dimensional and two-dimensional simulations. This comparison clearly shows the need for full three-dimensional simulations of droplet and bubble deformation to capture the correct physical behavior. Copyright © 2009 John Wiley & Sons, Ltd. [source] ## MUSTA schemes for multi-dimensional hyperbolic systems: analysis and improvements INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDS, Issue 2 2005V. A. TitarevAbstract We develop and analyse an improved version of the multi-stage (MUSTA) approach to the construction of upwind Godunov-type fluxes whereby the solution of the Riemann problem, approximate or exact, is not required. The new MUSTA schemes improve upon the original schemes in terms of monotonicity properties, accuracy and stability in multiple space dimensions. We incorporate the MUSTA technology into the framework of finite-volume weighted essentially nonoscillatory schemes as applied to the Euler equations of compressible gas dynamics. The results demonstrate that our new schemes are good alternatives to current centred methods and to conventional upwind methods as applied to complicated hyperbolic systems for which the solution of the Riemann problem is costly or unknown. Copyright © 2005 John Wiley & Sons, Ltd. [source] ## Splitting methods for high order solution of the incompressible Navier,Stokes equations in 3D INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDS, Issue 10-11 2005Arnim BrügerAbstract The incompressible Navier,Stokes equations are discretized in space by a hybrid method and integrated in time by the method of lines. The solution is determined on a staggered curvilinear grid in two space dimensions and by a Fourier expansion in the third dimension. The space derivatives are approximated by a compact finite difference scheme of fourth-order on the grid. The solution is advanced in time by a semi-implicit method. In each time step, systems of linear equations have to be solved for the velocity and the pressure. The iterations are split into one outer iteration and three inner iterations. The accuracy and efficiency of the method are demonstrated in a numerical experiment with rotated Poiseuille flow perturbed by Orr,Sommerfeld modes in a channel. Copyright © 2005 John Wiley & Sons, Ltd. [source] ## Non-oscillatory relaxation methods for the shallow-water equations in one and two space dimensions INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDS, Issue 5 2004Mohammed SeaïdAbstract In this paper, a new family of high-order relaxation methods is constructed. These methods combine general higher-order reconstruction for spatial discretization and higher order implicit-explicit schemes or TVD Runge,Kutta schemes for time integration of relaxing systems. The new methods retain all the attractive features of classical relaxation schemes such as neither Riemann solvers nor characteristic decomposition are needed. Numerical experiments with the shallow-water equations in both one and two space dimensions on flat and non-flat topography demonstrate the high resolution and the ability of our relaxation schemes to better resolve the solution in the presence of shocks and dry areas without using either Riemann solvers or front tracking techniques. Copyright © 2004 John Wiley & Sons, Ltd. [source] ## Numerical simulation of three-dimensional free surface flows INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDS, Issue 7 2003V. MaronnierAbstract A numerical model is presented for the simulation of complex fluid flows with free surfaces in three space dimensions. The model described in Maronnier et al. (J. Comput. Phys. 1999; 155(2) : 439) is extended to three dimensional situations. The mathematical formulation of the model is similar to that of the volume of fluid (VOF) method, but the numerical procedures are different. A splitting method is used for the time discretization. At each time step, two advection problems,one for the predicted velocity field and the other for the volume fraction of liquid,are to be solved. Then, a generalized Stokes problem is solved and the velocity field is corrected. Two different grids are used for the space discretization. The two advection problems are solved on a fixed, structured grid made out of small cubic cells, using a forward characteristic method. The generalized Stokes problem is solved using continuous, piecewise linear stabilized finite elements on a fixed, unstructured mesh of tetrahedrons. The three-dimensional implementation is discussed. Efficient postprocessing algorithms enhance the quality of the numerical solution. A hierarchical data structure reduces memory requirements. Numerical results are presented for complex geometries arising in mold filling. Copyright © 2003 John Wiley & Sons, Ltd. [source] ## Equilibrium real gas computations using Marquina's scheme INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDS, Issue 3 2003Youssef StiribaAbstract Marquina's approximate Riemann solver for the compressible Euler equations for gas dynamics is generalized to an arbitrary equilibrium equation of state. Applications of this solver to some test problems in one and two space dimensions show the desired accuracy and robustness. Copyright © 2003 John Wiley & Sons, Ltd. [source] ## Numerical analysis of interfacial two-dimensional Stokes flow with discontinuous viscosity and variable surface tension INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDS, Issue 5 2001Zhilin LiAbstract A fluid model of the incompressible Stokes equations in two space dimensions is used to simulate the motion of a droplet boundary separating two fluids with unequal viscosity and variable surface tension. Our theoretical analysis leads to decoupled jump conditions that are used in constructing the numerical algorithm. Numerical results agree with others in the literature and include some new findings that may apply to processes similar to cell cleavage. The method developed here accurately preserves area for our test problems. Some interesting observations are obtained with different choices of the parameters. Copyright © 2001 John Wiley & Sons, Ltd. [source] ## On the domain dependence of solutions to the Navier,Stokes equations of a two-dimensional compressible flow MATHEMATICAL METHODS IN THE APPLIED SCIENCES, Issue 18 2009Fei JiangAbstract We consider the Navier,Stokes equations for compressible, barotropic flow in two space dimensions, with pressure satisfying p(,)=a,logd(,) for large ,, here d>1 and a>0. After introducing useful tools from the theory of Orlicz spaces, we prove a compactness result for the solution set of the equations with respect to the variation of the underlying bounded spatial domain. Especially, we get a general existence theorem for the system in question with no restrictions on smoothness of the bounded spatial domain. Copyright © 2009 John Wiley & Sons, Ltd. [source] ## A fourth-order parabolic equation in two space dimensions MATHEMATICAL METHODS IN THE APPLIED SCIENCES, Issue 15 2007Changchun LiuAbstract In this paper, we consider an initial-boundary problem for a fourth-order nonlinear parabolic equations. The problem as a model arises in epitaxial growth of nanoscale thin films. Based on the Lp type estimates and Schauder type estimates, we prove the global existence of classical solutions for the problem in two space dimensions. Copyright © 2007 John Wiley & Sons, Ltd. [source] ## On the existence of solutions to the Navier,Stokes,Poisson equations of a two-dimensional compressible flow MATHEMATICAL METHODS IN THE APPLIED SCIENCES, Issue 3 2007Yinghui ZhangAbstract In this paper, we consider the Navier,Stokes,Poisson equations for compressible, barotropic flow in two space dimensions. We introduce useful tools from the theory of Orlicz spaces. Then we prove the existence of globally defined finite energy weak solutions for the pressure satisfying p(,)=a,logd (,) for large ,. Here d>1 and a>0. Copyright © 2006 John Wiley & Sons, Ltd. [source] ## A posteriori error estimation for convection dominated problems on anisotropic meshes MATHEMATICAL METHODS IN THE APPLIED SCIENCES, Issue 7 2003Gerd KunertAbstract A singularly perturbed convection,diffusion problem in two and three space dimensions is discretized using the streamline upwind Petrov Galerkin (SUPG) variant of the finite element method. The dominant convection frequently gives rise to solutions with layers; hence anisotropic finite elements can be applied advantageously. The main focus is on a posteriori energy norm error estimation that is robust in the perturbation parameter and with respect to the mesh anisotropy. A residual error estimator and a local problem error estimator are proposed and investigated. The analysis reveals that the upper error bound depends on the alignment of the anisotropies of the mesh and of the solution. Hence reliable error estimation is possible for suitable anisotropic meshes. The lower error bound depends on the problem data via a local mesh Peclet number. Thus efficient error estimation is achieved for small mesh Peclet numbers. Altogether, error estimation approaches for isotropic meshes are successfully extended to anisotropic elements. Several numerical experiments support the analysis. Copyright © 2003 John Wiley & Sons, Ltd. [source] ## On the existence of solutions to the Navier,Stokes equations of a two-dimensional compressible flow MATHEMATICAL METHODS IN THE APPLIED SCIENCES, Issue 6 2003Radek ErbanWe consider the Navier,Stokes equations for compressible, barotropic flow in two space dimensions. We introduce useful tools from the theory of Orlicz spaces. Then we prove the existence of globally defined finite energy weak solutions for the pressure satisfying p(,) = a,logd(,) for large ,. Here d>1 and a > 0. Copyright © 2003 John Wiley & Sons, Ltd. [source] ## A fully discrete nonlinear Galerkin method for the 3D Navier,Stokes equations NUMERICAL METHODS FOR PARTIAL DIFFERENTIAL EQUATIONS, Issue 3 2008J.-L. GuermondAbstract The purpose of this paper is twofold: (i) We show that the Fourier-based Nonlinear Galerkin Method (NLGM) constructs suitable weak solutions to the periodic Navier,Stokes equations in three space dimensions provided the large scale/small scale cutoff is appropriately chosen. (ii) If smoothness is assumed, NLGM always outperforms the Galerkin method by a factor equal to 1 in the convergence order of the H1 -norm for the velocity and the L2 -norm for the pressure. This is a purely linear superconvergence effect resulting from standard elliptic regularity and holds independently of the nature of the boundary conditions (whether periodicity or no-slip BC is enforced). © 2007 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2008 [source] ## A new class of stabilized mesh-free finite elements for the approximation of the Stokes problem NUMERICAL METHODS FOR PARTIAL DIFFERENTIAL EQUATIONS, Issue 5 2004V. V. K. Srinivas KumarAbstract Previously, we solved the Stokes problem using a new linear - constant stabilized mesh-free finite element based on linear Weighted Extended B - splines (WEB-splines) as shape functions for the velocity approximation and constant extended B-splines for the pressure (Kumar et al., 2002). In this article we derive another linear-constant element that uses the Haar wavelets for the pressure approximation and a quadratic - linear element that uses quadrilateral bubble functions for the enrichment of the velocity approximation space. The inf-sup condition or Ladyshenskaya-Babus,ka-Brezzi (LBB) condition is verified for both the elements. The main advantage of these new elements over standard finite elements is that they use regular grids instead of irregular partitions of domain, thus eliminating the difficult and time consuming pre-processing step. Convergence and condition number estimates are derived. Numerical experiments in two space dimensions confirm the theoretical predictions. © 2004 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2004. [source] ## An upwind finite volume scheme and its maximum-principle-preserving ADI splitting for unsteady-state advection-diffusion equations NUMERICAL METHODS FOR PARTIAL DIFFERENTIAL EQUATIONS, Issue 2 2003Hong WangAbstract We develop an upwind finite volume (UFV) scheme for unsteady-state advection-diffusion partial differential equations (PDEs) in multiple space dimensions. We apply an alternating direction implicit (ADI) splitting technique to accelerate the solution process of the numerical scheme. We investigate and analyze the reason why the conventional ADI splitting does not satisfy maximum principle in the context of advection-diffusion PDEs. Based on the analysis, we propose a new ADI splitting of the upwind finite volume scheme, the alternating-direction implicit, upwind finite volume (ADFV) scheme. We prove that both UFV and ADFV schemes satisfy maximum principle and are unconditionally stable. We also derive their error estimates. Numerical results are presented to observe the performance of these schemes. © 2003 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 19: 211,226, 2003 [source] |