Interval Study of Convergence in the Solution of 1d Burgers by Least Squares Finite Element Method (lsfem) + Newton Linearization

This paper aims to apply the least squares finite element method (LSFEM) in conjunction with the linearization technique known as Newton method. For this, the application chosen was the traditional 1D Burgers equation. From two numerical applications and an error analysis based on knowledge of the exact solution, will be possible to present convergence ranges for this proposal.


INTRODUCTION
The vast majority of physical problems is governed, or may be represented by Partial Differential Equations.Some mathematical methods are capable of producing analytical solutions of physical problems, more precisely of heat and mass transfer problems (Arpaci, 1966;Bejan, 1996;Carslaw and Jaeger, 1986), but only of some, and problems very simplified.So it is taken as a fundamental tool for solving heat and mass transfer problems the numerical methods.For decades, numerical methods have been used to solve such problems, among them stand out the finite difference method (Smith, 1971), finite volume (Chung, 2002) and finite element (Lewis et al., 2004;Donea and Huerta, 2003;Reddy, 1993).
Since the start of the 50's with Turner et al. (1956), Clough (1960), Argyris (1963), Zienkiewicz and Cheung (1965), Oden and Wellford (1972), the finite element method has been used with great success in various branches of engineering.
In Donea and Quartapelle (1992) the authors present a finite element method to solve transient problems governed by linear or nonlinear equations with dominant advective terms.The first is the Generalized Galerkin Method, which provides excellent results due to a correct relationship between the spatial and temporal variations expressed by the theory of characteristics.Beyond that show that the least square method used shows the simplicity of the Taylor-Galerkin method and the unconditional stability of methods of characteristics, however, its accuracy is committed to numbers Courant larger than unity.
It is important to highlight the work of Burrel et al. (1995) where the authors present the numerical solution via LSFEM of a pollutant transport case in a onedimensional domain and the work of Bramble et al. (1998) which introduce and analyze two least squares method for elliptic differential equations of second-order with mixed boundary conditions.Vujicic and Brown (2004) show numerical solution of a three-dimensional transient heat conduction case, in which several time discretization methods have been tested, among which especially the method value β where β is admitted to 0, 0.5, 0.75 to 1 besides using of LSFEM.Whereas Romão and Moura (2012) discuss how numerically solving the partial differential equation modeling convection-diffusion-reaction/heat generation in 3D domain with variable coefficients using the Galerkin method and the LSFEM.
In a new paper, Romão et al. (2011) again compare the results obtained from the two of the FEM variants: LSFEM and Galerkin method when applied to the Poisson equation and the Helmholtz equation.The analysis is made in Poisson's equation applied to diffusion in solids and in the Helmholtz equation applied to diffusion in solids with generation.As well as at work Romão and Moura (2012), the L 2 and L ∞ errors were also calculated and the values for each variant of the FEM, compared between themselves in accordance with the mesh refinement, and also compared with the error of the solutions to the same problems obtained from the finite difference method (FDM).In this work, the FEM was more appropriate not only for its greater accuracy, but also for its versatility when dealing with multi-connected and irregular domains.Comparing variants of the FEM, again the GFEM is most useful when one wants only obtain the solution to T(x,y,z), to be efficient and present a low computational cost, and LSFEM, is the most indicated for the calculation of the first derivative T. The accuracy in calculating the first derivative of T using LSFEM, in this study, gets to be three orders higher than the accuracy of results obtained from GFEM.The obtaining of the value of the first derivative of T is important in heat transfer studies, since it allows the analysis of the heat flow at any point of the domain.
The linearization technique used in this study is discussed by Campos et al. (2014) for solving a nonlinear transient diffusive-convective problem in three dimensions using FDM with order high accuracy.In this work, the authors applied the Crank-Nicolson technique to discretize the time.The technique is derivatives from the weighted average over time of the independent variable in two consecutive time steps by linear interpolation of the variable values, where the coefficients of these derivatives value is 0.5.To linearize the convective terms, the Newton method is applied, since it is a simple technique that optimizes the calculation because it does not require an iterative linearization in each step of time, decreasing your computing time.The numerical solution obtained was using a regular mesh and compared with the exact solution of equation.The L ∞ errors were calculated for each refinement made in the mesh and a table was presented, containing the computational time and the convergence rate.Presented in tables, conform mesh is refined, the more accurate the result becomes, because the oscillations of numerical solution are softened.
The same discretization time and linearization techniques used in Campos et al. (2014) are used in the work done by Cruz et al. (2014).This work solves the Burgers equation in one dimension using finite difference method in four types of applications.The results were compared with the exact solutions and with the results obtained into other works.Again, the techniques present satisfactory results making the solution of the problem more accurate.
The main objective of this work is to apply the finite element method in LSFEM variant united with the Newton method for linearization of the nonlinear term of the 1D Burgers equation and solve it numerically.

METHODOLOGY Model equation
Introduced here is a numerical study of the partial differential equation of the flow phenomenon in one direction without pressure gradient (equation Burgers 1D) defined in the domain Θ = Ξ ⊗ Ω, Ξ ⊂ ℜ, Ω ⊂ ℜ, in which Ξ and Ω are limited and closed domains, described as: In which υ is the cinematic viscosity (m/s 2 ) and u = u(x,t) is the flow field in the x direction.

Weighted residual method
In this work, the aim is to use the weighted residual method to obtain an approximate solution to the differential Equation (1).For it will be used functions attempts type: (2) So it can be defined an R residue as: Soon after, it introduces a set of weighting functions vi (i = 1, 2,..., Nnodes) and defines an inner product (R,vi).Then it is determined that the inner product is zero: Which is equivalent to force the error approximation of the differential equation is equal to zero on average.There are several ways to choose weighting functions vi.This work will use the least squares method.

Least squares method
To obtain the approximate solution of the problem (1) we will use the least squares method, for which basic idea is to determine u e ∈ V e for minimizing the integral of the square of the residue defined by Equation (3).To this end, a quadratic functional is defined: For all 1 u ∈ H 1 , in which H 1 is the Hilbert space of order 1.
The condition necessary so that 1 u ∈ V be a minimizer of the functional I in Equation ( 5) is that in first variation of 1 Note that when comparing the Equations ( 4) and ( 6) with that for the least squares method the weighting function is taken as the first variation of the residue.
The algebraic system generated by the formulation by the least squares method is symmetric and positive definite for any values of coefficients of Equation (1) in accordance with Jiang (1998).After formulation the finite element method, a linear system is generated, and from a computational code, the results for the velocity fields for each step time are found.Just for facility, we will use u instead of u1.

FORMULATION BY LEAST SQUARES METHOD
Rearranging the Equation (1), separating the derivatives in time and space, the following is obtained: Here, we use the Newton method (Jiang, 1998) to linearize the nonlinear term, as follows: Substituting Equation (8) into Equation ( 7) is obtained: Then, Equation ( 9) discretized the transient term, which in this work used Crank-Nicolson method resulting in the following equation: Simplifying and rearranging the equation ( 10), leads to: 11) the following residual equation can be written: and taking an approximation in the form: Where i N is the interpolation functions, obtaining the expression: Before using the results presented in Equation ( 6) it is necessary to define the first variation of the residue (Equation 14) is as follows: And thus, substituting the Equations ( 14), ( 15) and ( 16) into Equation ( 6), the following is obtained: Starting from the hypothesis that , it will be possible generate the following matrix vetor system: Considering our mesh composed of three elements called nodes m, n and p (with equal spacing between these nodes) in each element, and the following weighting functions: will be constructed a matrix vetor system of each element as follows:

NUMERICAL APPLICATIONS
From the matrix vetor system presented in Equation ( 21) is constructed for all the elements of the computational mesh an overall linear system from a computational code written in Fortran language.To solve the global linear system in each step of time is used the Gauss-Seidel method with stop error equal to 10 -6 . To analyze the efficiency of the proposed formulation presents two applications with exact solution for comparison.

Application 1
In this application, considering Re = 1, was used the exact solution Cruz et al., 2014) for comparison with the numerical results.After several refinements in space (x) and in time (t), for this application the proposed formulation present a clear convergence range.As can be seen in Tables 1 and 2, the numerical solution showed an efficiency of at less 10 -2 in mesh where the relation t/(x) 2 is between 0.25 and 0.45 (aproximately).It is important to note that several others combinations between x and t were tested validating the found interval, however these results were not presented because of table size.

Conclusions
The LSFEM proved extremely effective in solving the What should be highlighted is that the Newton method proposed here is only an iterative step in each time step, and this has led to this convergence range, but a In this work since it is a one-dimensional problem in its construction of the computational code a very important feature of problems solved with LSFEM was not   highlighted: Its formulation creates a positive-definite and symmetric matrix, which enables the use of a method of conjugated gradients which has a more significant convergence than the Gauss-Seidel method and the possibility of economy in the storage of non-zero coefficients to have a symmetric matrix.The authors believe these characteristics indicate that the LSFEM + Newton method is a good option for solution of nonlinear transient problems and in areas two and three dimensional.

Table 1 .
Several results of the L norm for different refinements.

Table 2 .
Different values for relation t/(x) 2 for different refinements.
proposal for future work is to use the LSFEM with the Newton method with an iterative process at each time step and analyze how it behaves this convergence range.

Table 3 .
Several results of the L norma for different refinements.

Table 4 .
Different values for relation t/(x) 2 for different refinements.