A posteriori error estimates of discontinuous Galerkin methods for the Signorini problem

A reliable and efficient a posteriori error estimator is derived for a class of discontinuous Galerkin (DG) methods for the Signorini problem. A common property shared by many DG methods leads to a unified error analysis with the help of a constraint preserving enriching map. The error estimator of DG methods is comparable with the error estimator of the conforming methods. Numerical experiments illustrate the performance of the error estimator. (C) 2015 Elsevier B.V. All rights reserved.

Singular value decomposition based computationally efficient algorithm for rapid dynamic near-infrared diffuse optical tomography

Purpose: A computationally efficient algorithm (linear iterative type) based on singular value decomposition (SVD) of the Jacobian has been developed that can be used in rapid dynamic near-infrared (NIR) diffuse optical tomography. Methods: Numerical and experimental studies have been conducted to prove the computational efficacy of this SVD-based algorithm over conventional optical image reconstruction algorithms. Results: These studies indicate that the performance of linear iterative algorithms in terms of contrast recovery (quantitation of optical images) is better compared to nonlinear iterative (conventional) algorithms, provided the initial guess is close to the actual solution. The nonlinear algorithms can provide better quality images compared to the linear iterative type algorithms. Moreover, the analytical and numerical equivalence of the SVD-based algorithm to linear iterative algorithms was also established as a part of this work. It is also demonstrated that the SVD-based image reconstruction typically requires O(NN2) operations per iteration, as contrasted with linear and nonlinear iterative methods that, respectively, requir O(NN3) and O(NN6) operations, with ``NN'' being the number of unknown parameters in the optical image reconstruction procedure. Conclusions: This SVD-based computationally efficient algorithm can make the integration of image reconstruction procedure with the data acquisition feasible, in turn making the rapid dynamic NIR tomography viable in the clinic to continuously monitor hemodynamic changes in the tissue pathophysiology.

Some Methods for Summarizing Survivorship Data in Nonstandard Situations

One difficulty in summarising biological survivorship data is that the hazard rates are often neither constant nor increasing with time or decreasing with time in the entire life span. The promising Weibull model does not work here. The paper demonstrates how bath tub shaped quadratic models may be used in such a case. Further, sometimes due to a paucity of data actual lifetimes are not as certainable. It is shown how a concept from queuing theory namely first in first out (FIFO) can be profitably used here. Another nonstandard situation considered is one in which lifespan of the individual entity is too long compared to duration of the experiment. This situation is dealt with, by using ancilliary information. In each case the methodology is illustrated with numerical examples.

Some direct numerical approaches to the solution of the singular nonlinear transonic integral equation

The nonlinear singular integral equation of transonic flow is examined, noting that standard numerical techniques are not applicable in solving it. The difficulties in approximating the integral term in this expression were solved by special methods mitigating the inaccuracies caused by standard approximations. It was shown how the infinite domain of integration can be reduced to a finite one; numerical results were plotted demonstrating that the methods proposed here improve accuracy and computational economy.

Numerical studies on quasilinear and linear elliptic equations

We explore here the acceleration of convergence of iterative methods for the solution of a class of quasilinear and linear algebraic equations. The specific systems are the finite difference form of the Navier-Stokes equations and the energy equation for recirculating flows. The acceleration procedures considered are: the successive over relaxation scheme; several implicit methods; and a second-order procedure. A new implicit method—the alternating direction line iterative method—is proposed in this paper. The method combines the advantages of the line successive over relaxation and alternating direction implicit methods. The various methods are tested for their computational economy and accuracy on a typical recirculating flow situation. The numerical experiments show that the alternating direction line iterative method is the most economical method of solving the Navier-Stokes equations for all Reynolds numbers in the laminar regime. The usual ADI method is shown to be not so attractive for large Reynolds numbers because of the loss of diagonal dominance. This loss can however be restored by a suitable choice of the relaxation parameter, but at the cost of accuracy. The accuracy of the new procedure is comparable to that of the well-tested successive overrelaxation method and to the available results in the literature. The second-order procedure turns out to be the most efficient method for the solution of the linear energy equation.

Numerical integration over arbitrary polygonal domains based on Schwarz-Christoffel conformal mapping

This paper presents a new numerical integration technique oil arbitrary polygonal domains. The polygonal domain is mapped conformally to the unit disk using Schwarz-Christoffel mapping and a midpoint quadrature rule defined oil this unit disk is used. This method eliminates the need for a two-level isoparametric mapping Usually required. Moreover, the positivity of the Jacobian is guaranteed. Numerical results presented for a few benchmark problems in the context of polygonal finite elements show that the proposed method yields accurate results.

Shortcomings of discontinuous-pressure finite element methods on a class of transient problems

Past studies that have compared LBB stable discontinuous- and continuous-pressure finite element formulations on a variety of problems have concluded that both methods yield Solutions of comparable accuracy, and that the choice of interpolation is dictated by which of the two is more efficient. In this work, we show that using discontinuous-pressure interpolations can yield inaccurate solutions at large times on a class of transient problems, while the continuous-pressure formulation yields solutions that are in good agreement with the analytical Solution.

Some nonstandard error analysis of discontinuous Galerkin methods for elliptic problems

An a priori error analysis of discontinuous Galerkin methods for a general elliptic problem is derived under a mild elliptic regularity assumption on the solution. This is accomplished by using some techniques from a posteriori error analysis. The model problem is assumed to satisfy a GAyenrding type inequality. Optimal order L (2) norm a priori error estimates are derived for an adjoint consistent interior penalty method.

Comparative study of different formulations and solution methods for two phase flow in saturated porous media

Six models (Simulators) are formulated and developed with all possible combinations of pressure and saturation of the phases as primary variables. A comparative study between six simulators with two numerical methods, conventional simultaneous and modified sequential methods are carried out. The results of the numerical models are compared with the laboratory experimental results to study the accuracy of the model especially in heterogeneous porous media. From the study it is observed that the simulator using pressure and saturation of the wetting fluid (PW, SW formulation) is the best among the models tested. Many simulators with nonwetting phase as one of the primary variables did not converge when used along with simultaneous method. Based on simulator 1 (PW, SW formulation), a comparison of different solution methods such as simultaneous method, modified sequential and adaptive solution modified sequential method are carried out on 4 test problems including heterogeneous and randomly heterogeneous problems. It is found that the modified sequential and adaptive solution modified sequential methods could save the memory by half and as also the CPU time required by these methods is very less when compared with that using simultaneous method. It is also found that the simulator with PNW and PW as the primary variable which had problem of convergence using the simultaneous method, converged using both the modified sequential method and also using adaptive solution modified sequential method. The present study indicates that pressure and saturation formulation along with adaptive solution modified sequential method is the best among the different simulators and methods tested.

A characteristic based numerical method with tracking for nonlinear wave equations

Many physical problems can be modeled by scalar, first-order, nonlinear, hyperbolic, partial differential equations (PDEs). The solutions to these PDEs often contain shock and rarefaction waves, where the solution becomes discontinuous or has a discontinuous derivative. One can encounter difficulties using traditional finite difference methods to solve these equations. In this paper, we introduce a numerical method for solving first-order scalar wave equations. The method involves solving ordinary differential equations (ODEs) to advance the solution along the characteristics and to propagate the characteristics in time. Shocks are created when characteristics cross, and the shocks are then propagated by applying analytical jump conditions. New characteristics are inserted in spreading rarefaction fans. New characteristics are also inserted when values on adjacent characteristics lie on opposite sides of an inflection point of a nonconvex flux function, Solutions along characteristics are propagated using a standard fourth-order Runge-Kutta ODE solver. Shocks waves are kept perfectly sharp. In addition, shock locations and velocities are determined without analyzing smeared profiles or taking numerical derivatives. In order to test the numerical method, we study analytically a particular class of nonlinear hyperbolic PDEs, deriving closed form solutions for certain special initial data. We also find bounded, smooth, self-similar solutions using group theoretic methods. The numerical method is validated against these analytical results. In addition, we compare the errors in our method with those using the Lax-Wendroff method for both convex and nonconvex flux functions. Finally, we apply the method to solve a PDE with a convex flux function describing the development of a thin liquid film on a horizontally rotating disk and a PDE with a nonconvex flux function, arising in a problem concerning flow in an underground reservoir.

{C}^0$ penalty methods for the fully nonlinear Monge-Ampère equation

In this paper, we develop and analyze C(0) penalty methods for the fully nonlinear Monge-Ampere equation det(D(2)u) = f in two dimensions. The key idea in designing our methods is to build discretizations such that the resulting discrete linearizations are symmetric, stable, and consistent with the continuous linearization. We are then able to show the well-posedness of the penalty method as well as quasi-optimal error estimates using the Banach fixed-point theorem as our main tool. Numerical experiments are presented which support the theoretical results.

A family of Runge-Kutta based explicit methods for rotational dynamics

The present paper develops a family of explicit algorithms for rotational dynamics and presents their comparison with several existing methods. For rotational motion the configuration space is a non-linear manifold, not a Euclidean vector space. As a consequence the rotation vector and its time derivatives correspond to different tangent spaces of rotation manifold at different time instants. This renders the usual integration algorithms for Euclidean space inapplicable for rotation. In the present algorithms this problem is circumvented by relating the equation of motion to a particular tangent space. It has been accomplished with the help of already existing relation between rotation increments which belongs to two different tangent spaces. The suggested method could in principle make any integration algorithm on Euclidean space, applicable to rotation. However, the present paper is restricted only within explicit Runge-Kutta enabled to handle rotation. The algorithms developed here are explicit and hence computationally cheaper than implicit methods. Moreover, they appear to have much higher local accuracy and hence accurate in predicting any constants of motion for reasonably longer time. The numerical results for solutions as well as constants of motion, indicate superior performance by most of our algorithms, when compared to some of the currently known algorithms, namely ALGO-C1, STW, LIEMID[EA], MCG, SUBCYC-M.

Multigrid algorithms for symmetric discontinuous Galerkin methods on graded meshes

We study a class of symmetric discontinuous Galerkin methods on graded meshes. Optimal order error estimates are derived in both the energy norm and the L 2 norm, and we establish the uniform convergence of V-cycle, F-cycle and W-cycle multigrid algorithms for the resulting discrete problems. Numerical results that confirm the theoretical results are also presented.

A simple numerical method for three-dimensional analysis of simple expansion chamber mufflers of rectangular as well as circular cross-section with a stationary medium

Three-dimensional effects are a primary source of discrepancy between the measured values of automotive muffler performance and those predicted by the plane wave theory at higher frequencies. The basically exact method of (truncated) eigenfunction expansions for simple expansion chambers involves very complicated algebra, and the numerical finite element method requires large computation time and core storage. A simple numerical method is presented in this paper. It makes use of compatibility conditions for acoustic pressure and particle velocity at a number of equally spaced points in the planes of the junctions (or area discontinuities) to generate the required number of algebraic equations for evaluation of the relative amplitudes of the various modes (eigenfunctions), the total number of which is proportional to the area ratio. The method is demonstrated for evaluation of the four-pole parameters of rigid-walled, simple expansion chambers of rectangular as well as circular cross-section for the case of a stationary medium. Computed values of transmission loss are compared with those computed by means of the plane wave theory, in order to highlight the onset (cutting-on) of various higher order modes and the effect thereof on transmission loss of the muffler. These are also compared with predictions of the finite element methods (FEM) and the exact methods involving eigenfunction expansions, in order to demonstrate the accuracy of the simple method presented here.