Defect Correction Technique Research Articles

A Parameter-Robust Numerical Method Based on Defect-Correction Technique for Parabolic Singular Perturbation Problems with Discontinuous Convection Coefficient and Source

The paper presents a uniformly convergent finite difference method based on the defect correction technique to solve parabolic singular perturbation problems with discontinuous convection coefficient and source. The solution to the problem exhibits interior layer across the discontinuity and demonstrates turning point behavior. The simultaneous presence of perturbation parameters and discontinuity makes the problem stiff. A higher-order method is developed using an implicit difference scheme in time on a uniform mesh and a combination of the upwind difference method and the central difference scheme over an adaptive mesh in space. The method involves iteratively solving increasingly accurate discrete problems by computing and using the defect to correct the approximate solution. Parameter uniform error estimates show uniform convergence of first-order in time and second-order in space. Numerical experiments confirm the accuracy of the proposed scheme and support the theoretical analysis.

Improving Regularization Techniques for Incompressible Fluid Flows via Defect Correction

Abstract We propose and investigate two regularization models for fluid flows at higher Reynolds numbers. Both models are based on the reduced ADM regularization (RADM). One model, which we call DC-RADM (deferred correction for reduced approximate deconvolution model), aims to improve the temporal accuracy of the RADM. The second model, denoted by RADC (reduced approximate deconvolution with correction), is created with a more systemic approach. We treat the RADM regularization as a defect in approximating the true solution of the Navier–Stokes equations (NSE) and then correct for this defect, using the defect correction algorithm. Thus, the resulting RADC model can be viewed as a first member of the class that we call “LESC-reduced”, where one starts with a regularization that resembles a Large Eddy Simulation turbulence model and then improves it with a defect correction technique. Both models are investigated theoretically and numerically, and the RADC is shown to outperform the DC-RADM model both in terms of convergence rates and in terms of the quality of the produced solution.

Local defect-correction method based on multilevel discretization for Steklov eigenvalue problem

In this paper, we propose a local defect-correction method for solving the Steklov eigenvalue problem arising from the scalar second order positive definite partial differential equations based on the multilevel discretization. The objective is to avoid solving large-scale equations especially the large-scale Steklov eigenvalue problem whose computational cost increases exponentially. The proposed algorithm transforms the Steklov eigenvalue problem into a series of linear boundary value problems, which are defined in a multigrid space sequence, and a series of small-scale Steklov eigenvalue problems in a coarse correction space. Furthermore, we use the local defect-correction technique to divide the large-scale boundary value problems into small-scale subproblems. Through our proposed algorithm, we avoid solving large-scale Steklov eigenvalue problems. As a result, our proposed algorithm demonstrates significantly improved the solving efficiency. Additionally, we conduct numerical experiments and a rigorous theoretical analysis to verify the effectiveness of our proposed approach.

Evaluation of an effective and robust implicit time-integration numerical scheme for Navier-Stokes equations in a CFD solver for compressible flows

The present work describes the implementation of an implicit time-integration numerical scheme to solve viscous flows in an in-house CFD solver. The scheme is developed to calculate engineering problems involving compressible flows. This work extends the defect-correction technique for the 3D flow calculations, and all mathematical formulations are described. The CFD solver is based on the finite-volume method (FVM) to calculate the three-dimensional flow and can be applied to solve unstructured meshes. The current implementation uses the Flux-Difference Splitting method (FDS) developed by Roe combined with the MUSCL method and the Venkatakrishnan flux limiters to provide better accuracy of the numerical solutions. The implicit time-integration scheme was linearized applying the backward Euler method on the left-hand side (LHS) and a Newton-type linearization on the right-hand side (RHS) of the governing equations. The Jacobian matrix was computed analytically for the inviscid fluxes using the Roe fluxes, and for the viscous fluxes differentiating the conservative vector. Earlier work by Cavalca et al. (2018) showed the robustness and accuracy of this implicit solver to predict inviscid flows over the airfoil and into the supersonic nozzle. Finally, the Gauss-Seidel (GS) iterative method was applied to solve the resultant sparse and large system of equations. These numerical schemes and methods were applied to solve the laminar flow over a flat plate. Afterwards, the numerical solution was validated and verified with the exact Blasius solution. From the results, the numerical simulations exhibited superior robustness of the implicit-defect correction scheme when compared with the explicit scheme for compressible flows. All numerical particularities and their implementations are detailed in this paper.

The Local and Parallel Finite Element Scheme for Electric Structure Eigenvalue Problems

In this paper, an efficient multiscale finite element method via local defect-correction technique is developed. This method is used to solve the Schrödinger eigenvalue problem with three-dimensional domain. First, this paper considers a three-dimensional bounded spherical region, which is the truncation of a three-dimensional unbounded region. Using polar coordinate transformation, we successfully transform the three-dimensional problem into a series of one-dimensional eigenvalue problems. These one-dimensional eigenvalue problems also bring singularity. Second, using local refinement technique, we establish a new multiscale finite element discretization method. The scheme can correct the defects repeatedly on the local refinement grid, which can solve the singularity problem efficiently. Finally, the error estimates of eigenvalues and eigenfunctions are also proved. Numerical examples show that our numerical method can significantly improve the accuracy of eigenvalues.

Ocena funkcjonalna kończyn górnych po operacjach naprawczych wad wrodzonych łuku aorty u dzieci w zależności od unaczynienia tętniczego

Congenital heart defects are often accompanied by defects in great vessels, such as aortic coarctation with various forms of aortic arc hypoplasia, vascular rings and positional anomalies. Surgical treatment and the choice of optimal defect correction technique in the child may affect the restoration, maintenance or the arrest of blood supply by the aorta to arterial vessels given off by the aortic arch. Proper, closest physiological vascularisation of individual parts of the body and organs is necessary for proper development and maintenance of physiological function. The consequence of the use of various strategies for correcting congenital defects of the aorta and the use of available surgical techniques may be perfusion disorders, also in the field of vessels supplying upper limbs, most often the upper left limb. In this article, we have attempted to develop a universal strategy for assessing the development and function of the upper extremities in children. It consists of cardiac methods of cardiovascular evaluation and physiotherapeutic function tests of the upper limbs adapted to the age of patients. Confirmation of the long-term importance of maintaining or restoring the blood supply by the aorta on the basis of objective function tests of the upper extremities may be an additional argument while choosing the optimal surgical method in the youngest children.

Convergence Properties of Local Defect Correction Algorithm for the Boundary Element Method

Sometimes boundary value problems have isolated regions where the solution changes rapidly. Therefore, when solving numerically, one needs a fine grid to capture the high activity. The fine grid can be implemented as a composite coarse-fine grid or as a global fine grid. One cheaper way of obtaining the composite grid solution is the use of the local defect correction technique. The technique is an algorithm that combines a global coarse grid solution and a local fine grid solution in an iterative way to estimate the solution on the corresponding composite grid. The algorithm is relatively new and its convergence properties have not been studied for the boundary element method. In this paper the objective is to determine convergence properties of the algorithm for the boundary element method. First, we formulate the algorithm as a fixed point iterative scheme, which has also not been done before for the boundary element method, and then study the properties of the iteration matrix. Results show that we can always expect convergence. Therefore, the algorithm opens up a real alternative for application in the boundary element method for problems with localised regions of high activity.

Development and convergence analysis of an effective and robust implicit Euler solver for 3D unstructured grids

This paper reports the development and convergence analysis in steady-state of an effective and robust implicit finite-volume solver for compressible Euler equations on three-dimensional unstructured grids. A second-order upwind scheme (MUSCL) was employed based on Roe's approximate Flux-Difference Scheme (FDS) by using Venkatrakishnan flux limiters. The construction of the linear system for the implicit scheme was performed by applying the backward Euler on the left-hand side of the conservation equation and Newton-type linearization on the right-hand side. The Jacobian matrix that resulted from the linearization process was computed analytically using Roe flux terms. In this phase, the defect-correction technique was employed allowing effective time-dependent computations by an implicit time-integration scheme. In this approach, the flux integral on the right-hand side is computed based on a high-order of accuracy whilst the left-hand side the Jacobian is performed based on the low-order. The resulting sparse and large system of linear equations is solved by a sequential Gauss–Seidel iterative method. Simulations were performed and the developed implicit defect-correction solver was validated and verified. In addition, convergence analysis comparing the implicit solver and the explicit Runge–Kutta of 5-steps using Implicit Residual Smoothing (IRS) were performed showing the significant speed-up of the implicit solver over the explicit one. Simulations were performed for case studies to demonstrate the robustness of the developed implicit defect-correction solver in solving typical problems of aerodynamic involving transonic condition and shock wave captures for internal and external flows. Finally, the main particularities of the implicit scheme were investigated and discussed considering the simulation results, showing also its capacity to serve as an effective preconditioner (start-up method) to other implicit techniques.

Error estimate of a fully discrete defect correction finite element method for unsteady incompressible Magnetohydrodynamics equations

ABSTRACTIn this study, a fully discrete defect correction finite element method for the unsteady incompressible Magnetohydrodynamics (MHD) equations, which is leaded by combining the Back Euler time discretization with the two-step defect correction in space, is presented. It is a continuous work of our formal paper [Math Method Appl Sci. 2017. DOI:10.1002/mma.4296]. The defect correction method is an iterative improvement technique for increasing the accuracy of a numerical solution without applying a grid refinement. Firstly, the nonlinear MHD equation is solved with an artificial viscosity term. Then, the numerical solutions are improved on the same grid by a linearized defect-correction technique. Then, we introduce the numerical analysis including stability analysis and error analysis. The numerical analysis proves that our method is stable and has an optimal convergence rate. Some numerical results [see Math Method Appl Sci. 2017. DOI:10.1002/mma.4296] show that this method is highly efficient for the unsteady incompressible MHD problems.

Defect-deferred correction method for the two-domain convection-dominated convection–diffusion problem

We present a method for solving a fluid–fluid interaction problem (two convection-dominated convection–diffusion problems adjoined by an interface), which is a simplified version of the atmosphere–ocean coupling problem. The method resolves some of the issues that can be crucial to the fluid–fluid interaction problems: it is a partitioned time stepping method, yet it is of high order accuracy in both space and time (the two-step algorithm considered in this report provides second order accuracy); it allows for the usage of the legacy codes (which is a common requirement when resolving flows in complex geometries), yet it can be applied to the problems with very small viscosity/diffusion coefficients. This is achieved by combining the defect correction technique for increased spatial accuracy (and for resolving the issue of high convection-to-diffusion ratio) with the deferred correction in time (which allows for the usage of the computationally attractive partitioned scheme, yet the time accuracy is increased beyond the usual result of partitioned methods being only first order accurate) into the defect-deferred correction method (DDC). The results are readily extendable to the higher order accuracy cases by adding more correction steps. Both the theoretical results and the numerical tests provided demonstrate that the computed solution is unconditionally stable and the accuracy in both space and time is improved after the correction step.

Defect-correction finite element method based on Crank-Nicolson extrapolation scheme for the transient conduction-convection problem with high Reynolds number

A defect-correction finite element (FE) method is designed and analyzed for solving the two-dimensional (2D) transient conduction-convection problem at high Reynolds number. The method combines the merits of Crank-Nicolson (CN) extrapolation discretization and defect-correction scheme, which consists of solving a linearized problem with an added artificial viscosity term and then correcting the previous numerical solutions by a linearized defect-correction technique. The stability and optimal error estimate of the fully discrete scheme are derived. Finally, performance of the proposed method is investigated by numerical experiments.

Temperopareital Methyl Methacrylate Cranioplasty-A Preliminary Case Study And Technical Note

Temperoparietalcranioplasty are largely performed among all other varieties of craniolplasty and the role of Dental surgeon in these case is discussed in elaborate in restoring the defect with polymethyl methacrylate plate Post trauma cranial defects in 5 selected cases and their defect correction technique discussed in this paper.

Defect correction finite element method for the stationary incompressible Magnetohydrodynamics equation

In this study, we give the defect correction finite element method for the stationary incompressible MHD equation. Firstly, the nonlinear MHD equation is solved with an artificial viscosity term. Then, the numerical solution is improved on the same grid by a linearized defect-correction technique. Then, we give the numerical analysis including stability analysis and error analysis. The numerical analysis proves that our method is stable and has an optimal convergence rate. Then, we give some numerical results. From the numerical results, we can see that our method is efficient for solving the MHD equations.

Multilevel finite element discretizations based on local defect correction for nonsymmetric eigenvalue problems

Based on the work of Xu and Zhou (2000), we establish new three-level and multilevel finite element discretizations by local defect correction techniques. Theoretical analysis and numerical experiments show that the discretizations are simple and easy to implement, and can be used to solve nonsymmetric eigenvalue problems with non smooth eigenfunctions efficiently. We also discuss the local error estimates of finite element approximations; it is a new feature here that the estimates apply to the local domains containing corner points.

The multilevel mixed finite element discretizations based on local defect-correction for the Stokes eigenvalue problem

Abstract Based on the work of Xu and Zhou (2000), this paper combines mixed finite element method and the local defect-correction technique to establish new local and parallel multilevel discretization schemes for the Stokes eigenvalue problem. Theoretical analysis and numerical experiments show that the computational approach proposed in this paper is simple and easy to carry out in parallel, and can be used to solve singular Stokes eigenvalue problem efficiently. This paper also discusses local error estimates of mixed finite element approximations; it is a new feature here that the estimates apply to the local domains containing corner points.

Modified characteristics mixed defect-correction finite element method for the time-dependent Navier–Stokes problems

In this paper, a modified method of characteristics mixed defect-correction finite element method (MMOCMDCFEM) for the time-dependent Navier–Stokes problems, which is led by combining the characteristics time discretization with the two-step defect-correction in space, is presented. In this method, the hyperbolic part (the temporal and advection term) are treated by a characteristic tracking scheme. Then, we solve the equations with an added artificial viscosity term and correct these solutions by using a defect-correction technique. The theory analysis shows that this method has a good convergence property. In order to show the efficiency of the MMOCMDCFEM, we first present some numerical results of analytical solution problems. Then, we give some numerical results of lid-driven cavity flow with and . We also give some numerical results of flow around a cylinder problems, and compare the results with MMOCFEM’s. The numerical results show that this method is highly Efficient.

Steady flow past a sphere in an aligned magnetic field at high Reynolds numbers

The flow of steady, incompressible, viscous, electrically conducting fluid past a sphere in the presence of an uniform magnetic field parallel to the undisturbed flow is investigated using the finite difference method. The multigrid method with defect correction technique is used to achieve the second order accurate solution. The Hartmann number, M is used as the perturbation parameter. It is found that the increase of magnetic field decreases the wake length and increases the drag coefficient. The graphs of streamlines, vorticity lines, drag coefficient, surface pressure and surface vorticity are presented and discussed. Keywords: Navier-Stokes equations, MHD, Hartmann number, Multigrid method, Defect correctionInternational Journal of Engineering, Science and Technology, Vol. 6, No. 1, 2014, pp

Numerical solutions for steady viscous flow past a circular cylinder in an aligned magnetic field

The steady incompressible Magneto-Hydrodynamic (MHD) flow past a circular cylinder with an aligned magnetic field is studied for the Reynolds number (Re) up to 100, using the Hartmann number M, as the perturbation parameter. The multigrid method with defect correction technique is used to achieve the second order accurate solution of complete nonlinear Navier-Stokes equations. The magnetic Reynolds number is assumed to be small. It is observed that the volume of the separation bubble decreases and drag coefficient increases as M is increased. The graphs of streamlines, vorticity lines, surface pressure, surface vorticity and drag coefficient are presented and the effect of magnetic field is discussed.

Local and Parallel Finite Element Discretizations for Eigenvalue Problems

Based on the work of Xu and Zhou [Math. Comp., 69 (2000), pp. 881--909], this paper combines the local defect-correction technique and the shifted-inverse power method to establish new local and parallel finite element three-scale schemes for a class of eigenvalue problems. It is proved that with these schemes, the solution of an eigenvalue problem on a fine grid $\pi_{h}$ is reduced to the solution of an eigenvalue problem on a much coarser grid $\pi_{H}$, the solution of a linear algebraic system on a globally mesoscopic grid $\pi_{w}$, and the solutions of linear systems on several locally fine grids in parallel. The principle to determine the diameters of three different scale grids is given. Especially, this paper devises a new local and parallel finite element multiscale discretization scheme. Theoretical analysis and numerical experiments show that the computational approach proposed in this paper is simple and easy to carry out and can be used to solve singular eigenvalue problems efficiently.

An efficient higher order compact scheme to capture heat transfer solutions in spherical geometry

A higher order compact scheme (HOCS) is developed for solving energy equation in spherical polar coordinates for the flow past a sphere. The velocity components are obtained by solving the Navier–Stokes equations using HOCS in spherical geometry. The convection terms in the momentum and energy equations are handled in an effective manner so as to get fourth order accurate solutions. The higher order compact scheme is made efficient by combining with the coarse grid correction based multigrid method. In the absence of an exact solution, the order of accuracy of the results is illustrated. The results are compared with first order accurate upwind scheme, second order accurate defect correction technique(DC) and other recent numerical results of the literature. It is found that HOCS captures heat transfer solutions accurately when compared to other conventional schemes at higher values of Re and Pr. The angular variation of thermal boundary layer thickness is also presented.