Least-squares Spectral Element Research Articles

Fully discrete least-squares spectral element method for parabolic interface problems

In this article we propose fully discrete least-squares spectral element method for parabolic interface problems in R2. Crank–Nicolson scheme is used in time and higher order spectral elements are used in the spatial direction. This method is based on the nonconforming spectral element method proposed in Kishore Kumar and Naga Raju (2012). The proposed method is least-squares spectral element method. Nonconforming higher order spectral elements have been used. The jump in the solution and its normal derivative across the interface are enforced (in an appropriate Sobolev norm) in the minimizing functional. The method is second order accurate in time and exponentially accurate in spatial direction with p−version in L2(H1) norm. Numerical results are presented to show the efficiency of the proposed method.

Spectral Method and Spectral Element Method for Three Dimensional Linear Elliptic System: Analysis and Application

In this paper, a least-squares spectral method and a non-conforming least-squares spectral element method for three dimensional linear elliptic system will be presented. Differentiability estimates and the main stability theorem for the proposed method are proven. Using the regularity estimate and the proposed stability estimates, we introduce a suitable preconditioner and show that the condition number of the preconditioned system is O((ln W)^2) (where W is degree of polynomials). Then we establish the error estimates of the proposed method. Specific numerical results based on linear elasticity problem, fourth order problem and sixth order problem are discussed to reflect the efficiency of the proposed method.

Non-conforming least-squares spectral element method for Stokes equations on non-smooth domains

In this paper we present a discontinuous least-squares spectral element method for Stokes equations with primitive variable formulation on both smooth and non-smooth domains. We propose an exponentially accurate numerical scheme based on the stability estimates and implement it on different polygonal domains. Preconditioned conjugate gradient method is used to obtain the numerical solution.

Space-Time Coupled Least-Squares Spectral Element Methods for Parabolic Problems*

We study a space-time coupled least-squares spectral element method for parabolic initial boundary value problems using parallel computers. The method is spectrally accurate in both space and time. The spectral element functions are polynomials of degree q in the time variable and of degree p in each of the space variables and are non-conforming in both space and time. There is no need therefore to consider C0 element expansions at the inter element boundaries and consequently it is unnecessary to express the equation as an equivalent set of first-order equations by introducing an additional independent variable. Instead, we consider the jump in the function and its derivatives across inter-element boundaries in certain Sobolev norms. The normal equations obtained from the least-squares formulation can be solved by a preconditioned conjugate gradient method (PCGM) in matrix-free form. The method is implemented on a parallel computer using message-passing-interface (MPI). We take q proportional to p2 in the p-version of the method, whereas in the h-version. If the solution belongs to a certain Gevrey space, then the error decays exponentially in p in the p-version of the method and the error is in the h-version. Further, the number of iterations, needed to obtain the approximate solution using PCGM, is per time step. These estimates are confirmed by the computational results.

Performance of space-time coupled least-squares spectral element methods for parabolic problems

We study the performance of space-time coupled least-squares spectral element method (LSSEM) for parabolic initial boundary value problem (IBVP) for different values of element size h, the time step k, the degree q in the time variable and the degree p in each of the space variables. We divide the space domain into a number of shape regular quadrilaterals of size h and the time step k is proportional to h2. Each shape regular quadrilateral will be mapped to the square (−1,1) × (−1,1). In each square the solution will be defined as a polynomial of degree p for the space variables and degree q for the time variable. For the p version of the method h remains fixed and p increases, for the the h-version of the method p remains fixed and h decreases but for the hp-version of the method h decreases and at the same time p also increases. In this method, k is proportional to h2 (say k = ch2) and for the p-version of the method, q is proportional to p2(say q = c′p2). The performance of the method is discussed for different values of c and c′. The method we discussed here is spectral in both space and time and are non-conforming.

Thermal two-phase flow with a phase-field method

Phase field methods have become of great interest for the simulation of droplet and bubble dynamics, moving free interfaces and more recently phase change phenomena. One example is the Navier–Stokes–Korteweg (NSK) equations. With a particular focus on the van der Waals fluid, we present the numerical solution of the NSK system with the energy equation included. The least-squares spectral element formulation with a time-stepping procedure, a high-order continuity approximation and an element-by-element technique is implemented to provide a general and robust solver for the thermal NSK equations. A convergence analysis is conducted to verify our solver. Two numerical examples regarding phase transitions of a droplet and thermocapillary convection are provided to validate our solver.

Formula omitted] continuous [formula omitted]-adaptive least-squares spectral element method for phase-field models

To describe the interfacial dynamics between two phases using the phase-field method, the interfacial region needs to be close enough to a sharp interface so as to reproduce the correct physics. Due to the high gradients of the solution within the interfacial region and consequent high computational cost, the use of the phase-field method has been limited to the small-scale problems whose characteristic length is similar to the interfacial thickness. By using finer mesh at the interface and coarser mesh in the rest of computational domain, the phase-field methods can handle larger scale of problems with realistic interface thicknesses. In this work, a C1 continuous h-adaptive mesh refinement technique with the least-squares spectral element method is presented. It is applied to the Navier–Stokes-Cahn–Hilliard (NSCH) system and the isothermal Navier–Stokes–Korteweg (NSK) system. Hermite polynomials are used to give global differentiability in the approximated solution, and a space–time coupled formulation and the element-by-element technique are implemented. Two refinement strategies based on the solution gradient and the local error estimators are suggested, and they are compared in two numerical examples.

The least-squares spectral element method for phase-field models for isothermal fluid mixture

The phase-field approach has been regarded as a powerful method in numerically handling the interface dynamics in multiphase flow in several scientific and engineering applications. For an isothermal fluid mixture, the Navier–Stokes–Korteweg equation and the Navier–Stokes–Cahn–Hilliard equation have represented two major branches of the phase-field methods. We present a general discretization formulation for these two equations and conduct a comparison study of them. The formulation using a least-squares spectral element method is implemented by adopting a time-stepping procedure, a high-order continuity approximation and an element-by-element solver technique. To describe the same fluid mixtures by the isothermal Navier–Stokes–Korteweg and the Navier–Stokes–Cahn–Hilliard equations, we suggest a non-dimensionalization with the same dimensionless quantities. Numerical experiments are conducted to verify the spectral/hp least-squares formulation for the isothermal Navier–Stokes–Korteweg model. Besides, the equilibrium state of the van der Waals fluid model is calculated both analytically and numerically. Through spontaneous decomposition example, the isothermal Navier–Stokes–Korteweg system and the Navier–Stokes–Cahn–Hilliard system are compared in terms of the equilibrium pressure and the energy minimizing process. As a general example, the coalescence of two liquid droplets is studied with our solver for the isothermal Navier–Stokes–Korteweg system. The minimum discretization levels for space and time are investigated and a parametric study on Weber number is carried out.

Least-squares spectral element preconditioners for fourth order elliptic problems

In this paper, we propose preconditioners for the system of linear equations that arise from a discretization of fourth order elliptic problems in two and three dimensions (d=2,3) using spectral element methods. These preconditioners are constructed using separation of variables and can be diagonalized and hence easy to invert. For second order elliptic problems this technique has proven to be successful and performs better than other preconditioners in the framework of least-squares methods. We show that these preconditioners are spectrally equivalent to the quadratic forms by which we approximate them. Numerical results for the condition number reflects the effectiveness of the preconditioners.

Spectral Element Method for Parabolic Initial Value Problem with Non-Smooth Data: Analysis and Application

In this paper, a least-squares spectral element method for parabolic initial value problem for two space dimension on parallel computers is presented. The theory is also valid for three dimension. This method gives exponential accuracy in both space and time. The method is based on minimization of residuals in terms of the partial differential equation and initial condition, in different Sobolev norms, and a term which measures the jump in the function and its derivatives across inter-element boundaries in appropriate fractional Sobolev norms. Rigorous error estimates for this method are given. Some specific numerical examples are solved to show the efficiency of this method.

Spectral element method for three dimensional elliptic problems with smooth interfaces

In this paper we propose a least-squares spectral element method for three dimensional elliptic interface problems. The differentiability estimates and the main stability theorem, using non-conforming spectral element functions, are proven. The proposed method is free from any kind of first order reformulation. A suitable preconditioner is constructed with help of the regularity estimate and proposed stability estimates which is used to control the condition number. We show that these preconditioners are spectrally equivalent to the quadratic forms by which we approximate them. We obtain the error estimates which show the exponential accuracy of the method. Numerical results are obtained for both straight and curved interfaces to show the efficiency of the proposed method.

Exponentially Accurate Spectral Element Method for Fourth Order Elliptic Problems

In this paper, a fully non-conforming least-squares spectral element method for fourth order elliptic problems on smooth domains is presented. The proposed method works for a general fourth order elliptic operator with non-homogeneous data in two dimensions and gives exponentially accurate solutions. We derive differentiability estimates and prove our main stability estimate theorem using a non-conforming spectral element method. We then formulate a numerical scheme using a block diagonal preconditioner. Error estimates are also proven for the proposed method. We provide the computational complexity of our method and present results of numerical simulations that have been performed to validate the theory.

A Non-Conforming Least Squares Spectral Element Formulation for Oseen Equations with Applications to Navier-Stokes Equations

ABSTRACTIn this article, we propose a non-conforming exponentially accurate least-squares spectral element method for Oseen equations in primitive variable formulation that is applicable to both two- and three-dimensional domains. First-order reformulation is avoided, and the condition number is controlled by a suitable preconditioner for velocity components and pressure variable. A preconditioned conjugate gradient method is used to obtain the solution. Navier-Stokes equations in primitive variable formulation have been solved by solving a sequence of Oseen type iterations. For numerical test cases, similar order convergence has been achieved for all Reynolds number cases at the cost of higher iteration number for higher Reynolds number.

Exponentially accurate nonconforming least-squares spectral element method for elliptic problems on unbounded domain

Non-conforming approximation methods are becoming increasingly popular because of the potential to apply to multi-material and multi-model analysis for both bounded and unbounded domains. In this paper, we present a least-square approximation based method to solve the one or two dimensional elliptic problems on an unbounded domain. The method gives exponential accuracy and shows superior performance when compared to other numerical methods. Differentiability estimates and the main stability estimate theorem, using a non-conforming spectral element method, are also discussed. The exponential convergence rate of the proposed method is also shown through rigorous error estimate and specific numerical examples.

Least-squares spectral element method for three dimensional Stokes equations

In this paper we propose a non-conforming least squares spectral element method for Stokes equations on three dimensional domains. Any kind of first order transformation has been avoided by using a block diagonal preconditioner for velocity and pressure variables. Preconditioned conjugate gradient method is used to obtain the numerical solution. The method is proved to be exponentially accurate. Numerical results are provided to validate the proposed estimates.

Numerical Solution of Incompressible Cahn-Hilliard and Navier-Stokes System with Large Density and Viscosity Ratio Using the Least-Squares Spectral Element Method

Numerical Solution of Incompressible Cahn-Hilliard and Navier-Stokes System with Large Density and Viscosity Ratio Using the Least-Squares Spectral Element Method

NONCONFORMING SPECTRAL ELEMENT METHOD FOR ELASTICITY INTERFACE PROBLEMS

An exponentially accurate nonconforming spectral element method for elasticity systems with discontinuities in the coefficients and the flux across the interface is proposed in this paper. The method is least-squares spectral element method. The jump in the flux across the interface is incorporated (in appropriate Sobolev norm) in the functional to be minimized. The interface is resolved exactly using blending elements. The solution is obtained by the preconditioned conjugate gradient method. The numerical solution for different examples with discontinuous coefficients and non-homogeneous jump in the flux across the interface are presented to show the efficiency of the proposed method.

Least-squares spectral element solution of incompressible Navier–Stokes equations with adaptive refinement

Least-squares spectral element solution of steady, two-dimensional, incompressible flows are obtained by approximating velocity, pressure and vorticity variable set on Gauss–Lobatto–Legendre nodes. Constrained Approximation Method is used for h- and p-type nonconforming interfaces of quadrilateral elements. Adaptive solutions are obtained using a posteriori error estimates based on least squares functional and spectral coefficient. Effective use of p-refinement to overcome poor mass conservation drawback of least-squares formulation and successful use of h- and p-refinement together to solve problems with geometric singularities are demonstrated. Capabilities and limitations of the developed code are presented using Kovasznay flow, flow past a circular cylinder in a channel and backward facing step flow.

Simulation of Multi-Component Mass Diffusion Pellet Model Using Least Squares Spectral Element Method for Steam Methane Reforming Process

Mass based pellet model has been solved using least-squares formulation to describe the evolution of species composition, pressure, velocity, total concentration, mass diffusion flux in porous pellets for the steam methane reforming (SMR) process. The diffusion-reaction problems are computationally intensive, requiring efficient numerical methods for dealing with them. This paper presents formulation and algorithm of least-squares spectral element method (LS-SEM) for solving multicomponent mass diffusion pellet models. The mass diffusion flux is described according to the rigorous Maxwell Stefan model. The effectiveness factors have been calculated for the SMR process and compared with the literature data. The model evaluations revealed that the least-squares method is well suited for solving the multicomponent mass diffusion pellet models for the SMR process, achieving exponential convergence.

Mass and momentum conservation of the least-squares spectral collocation method for the Navier–Stokes equations

From the literature, it is known that the Least-Squares Spectral Element Method (LSSEM) for the stationary Stokes equations performs poorly with respect to mass conservation but compensates this lack by a superior conservation of momentum. Furthermore, it is known that the Least-Squares Spectral Collocation Method (LSSCM) leads to superior conservation of mass and momentum for the stationary Stokes equations. In the present paper, we consider mass and momentum conservation of the LSSCM for time-dependent Stokes and Navier–Stokes equations. We observe that the LSSCM leads to improved conservation of mass (and momentum) for these problems. Furthermore, the LSSCM leads to the well-known time-dependent profiles for the velocity and the pressure profiles. To obtain these results, we use only a few elements, each with high polynomial degree, avoid normal equations for solving the overdetermined linear systems of equations and introduce the Clenshaw–Curtis quadrature rule for imposing the average pressure to be zero. Furthermore, we combined the transformation of Gordon and Hall (transfinite mapping) with the least-squares spectral collocation scheme to discretize the internal flow problems.