THE ACCURATE ELEMENT METHOD: A NOVEL METHOD FOR INTEGRATING ORDINARY DIFFERENTIAL EQUATIONS

Transcription

1 AAS THE ACCURATE EEMENT METHOD: A NOVE METHOD FOR INTEGRATING ORDINARY DIFFERENTIA EQUATIONS Paul Cizmas and Maty Blumenfeld This paper presents the development of a new methodology for numerically solving ordinary differential equations. This methodology, which we call the accurate element method (AEM), breaks the connection between the solution accuracy and the number of unknowns. The differential equations are discretized by using finite elements, like in the finite element method (FEM). A key feature of the AEM is the methodology developed for eliminating unknowns inside the element by using the relations provided by the governing equations. The results of the AEM applied to a second-order ordinary differential equation (ODE) show that the degree of the local approximation function does not affect the number of unknowns used to discretize the differential equation. For a second-order ODE it is shown that the discretized solution has the same number of unknowns, whether the local approximation (or interpolation) function is a third-degree or a nineteen-degree polynomial. The implication of this result is that higher-degree interpolation functions can be used without increasing the number of unknowns. For a specified accuracy, the usage of elements with high-degree interpolation functions leads to a reduction in the necessary number of elements. Numerical results using the AEM, the shooting method and the relaxation method are presented and compared for several ODE boundary value problems. INTRODUCTION Numerical solution of differential equations is of paramount importance for most of the fields of engineering and science because it provides a way to investigate the governing equations of various physical phenomena. As the field of numerical methods for solving differential equations matured, some of the algorithms were preferred for various reasons and are currently part of the classical methodologies. If one restricts this discussion to ordinary differential equations, it is easy to identify the most popular methods of the moment: () Runge-Kutta methods, () Richardson extrapolation and its particular implementation as Bulirsch-Stoer method, and (3) predictor-corrector methods, 4 p.70. The most general and widely used methods for partial differential equations include the finite difference method, finite volume method and finite element method (FEM). The common Assistant Professor, Department of Aerospace Engineering, Texas A&M University, College Station, TX cizmas@tamu.edu. Professor Emeritus, Chair of Strength of Materials, Politehnica University Bucharest, Romania, E- mail:mblum@itcnet.ro.

2 goal of these methods is to approximate as well as possible the solution of the governing equations describing various physical phenomena. Several options are currently used to improve the accuracy of these methods: () increasing the degree of the polynomials that locally approximate the solution, that is, p-type refinement; () decreasing the size of the elements which discretize the computational domain in order to reduce the variation of the unknowns over the element, that is, h-type refinement; (3) using anisotropic elements, that is, having different orders of expansion for each spatial direction in an element; (4) using non-conforming elements with either h-type or p-type non-conformities; (5) using adaptive refinement strategies. High-order finite element methods 9, 6 and spectral element methods 7, 0, 3, 7 are commonly used to produce high accuracy solutions. Spectral element methods combine the high accuracy of spectral methods 6, and the flexibility of the finite element method. The Galerkin spectral element method, a subset of the spectral element method, is similar to the high-order finite element method in the sense that both are weighted residual techniques. A basic difference between the two methods is the fact that the weighting functions used in the Galerkin spectral element method are either Chebyshev or egendre polynomials. 5, 6 Anisotropic and non-conforming elements are also used to improve the solution accuracy of certain problems. The non-conforming elements can be either h-type or p-type non-conforming. Various special formulations have been developed for anisotropic and non-conforming elements, such as the constrained approximation method, 8,, 5 the mortar element method,, the FETI method, 9 the transition element method 3, and the iterative patching method. 0, 5 Another option for producing high accuracy solutions is to use adaptive refinement strategies. Adaptive methods have the ability to automatically adjust themselves to improve the solution of a problem. The adaptive methods are mainly used for: () problems for which a properly resolved initial mesh is not available; and () unsteady problems for which the initial mesh is not good enough at some instants of the solution. The local error estimation and the adaptive strategy are two important aspects of the adaptive methods. 7, 4, 8 For all the methods currently used to solve differential equations, the outcome of increasing solution accuracy is an increase in the number of unknowns. To minimize the number of unknowns, each method tries to optimize the relation between the degree of the approximation function and the size of the element, such as in the h p adaptive finite element or spectral element methods. This paper presents a new methodology for the numerical solution of differential equations which breaks the connection between the solution accuracy and the number of unknowns. Consequently, the number of unknowns corresponding to high-accuracy solutions obtained using high-order approximation polynomials will be equal to the number of unknowns corresponding to low-order approximation polynomials. This new methodology, which we call the accurate element method (AEM), will be applied herein to numerically solve linear and nonlinear ODEs. The results shown in this paper indicate that the degree of the local approximation function does not affect the number of unknowns used to discretize the differential equation. THE ACCURATE EEMENT METHOD (AEM) The AEM will be presented by comparing it to the FEM. The strategies of both the AEM and the FEM can be summarized by the following three steps: () discretization of the domain D on which the governing equations must be integrated; () local approximation of the solution of the governing equations, in which the information given by each element is concentrated at the nodes of the element; and (3) reconstruction of the domain D, obtained by writing nodal equations that bring together the information given by the elements adjacent to each node. The solution of the

3 system of equations obtained in this way represents the nodal unknowns. The first and third steps of the AEM are identical to those of the FEM. The two methods differ only in how the second step of their common strategy is accomplished. Note that the second step is crucial in obtaining an accurate solution of the governing equations. As will be shown herein, the AEM uses a much smaller number of elements than the FEM to obtain a similar accuracy. In step two, the local approximation of the solution in the FEM is done by using interpolation functions (IF). Usually these interpolation functions are low degree polynomials that lead to poor approximations. Consequently, the number of elements used in the FEM must be large in order to obtain a result with reasonable accuracy. The local approximation in the AEM will be done by high-degree polynomial functions. Because the functions used for local approximation are highdegree interpolation functions (HIF), only a small number of elements is needed to obtain an accurate solution in the AEM. The example presented in the next section shows that the accuracy obtained in the AEM by using two elements with a seventh-degree polynomial HIF is better than the accuracy obtained in the FEM by using 00 elements with a linear IF. The degree of the polynomials used in the AEM is usually higher than seven, since, as will be shown herein, the computational effort in the AEM is not affected by the degree of the polynomial. This is an important difference between the AEM and the FEM. High-degree polynomials may be used for the IF in the FEM, but this increases the number of unknowns of the discretized system of equations. In the AEM, the degree of the interpolation function used in the local approximation does not affect the number of unknowns of the discretized system of equations. This is an important advantage of the AEM compared to the finite element, finite volume or finite difference methods. To illustrate how the AEM can be implemented to solve ODEs, three cases will be presented: () a linear ODE with constants coefficients; () a linear ODE with variable coefficients; (3) a nonlinear ODE. All cases shown herein are second-order ODEs. AEM Applied to inear ODEs To introduce the methodology of the AEM, let us start by presenting a simple but yet meaningful case, the solution of a linear, second-order ordinary differential equation: d φ + cφ(x) + d(x) = 0, () where c is a constant and d(x) is a function of the independent variable x. et us consider the following boundary conditions φ := φ(x = 0) = 0 () φ := φ(x = ) = 0. et us consider a HIF that is a seventh-degree polynomial: φ(x) = C 0 + C x + C x + C 3 x C 7 x 7 (3) and let us write φ using a natural coordinate η = x/, where is the length of the element. Equation (3) can be written in vectorial form using the natural coordinate η: φ(η) = ηc, (4) 3

4 where η = η η... η 7 C = C 0 C C... C 7 7 T. The first goal is to write the local interpolation function φ(x) as a function of the values of φ(x) and its derivatives at nodal locations. The values of the function φ and its derivatives at both ends of the element can be written as: or φ () φ () 3 φ (3) φ () φ () 3 φ (3) = C (5) B = AC. (6) The notation i := φ i and φ (k) i B matrix can be rearranged as: 0 B = 0 + := dk φ i k, i, and k has been used in equation (5). The 0 0 φ () φ () φ () φ () and the inverse of the A matrix needs to be computed only once. The vector of coefficients C can be written using equations (6) and (7) as: C = A B = F 80 φ () + F 8 φ () + φ () F 8 φ () + 3 φ (3) F 83 φ (3) (8) φ (3) φ (3) (7) where F 80 = , F 8 = , F 8 = , 4

5 F 83 = By substituting the C vector from equation (8) into equation (4), the local approximation function φ(x) is written as a function of the nodal values of the function and its derivatives. et us derive a relation similar to but more general than the stiffness matrix relation. This and the variable derivatives φ (k) i, k. relation should provide a link between the variables i This relation, which is a generalization of the stiffness matrix relation, will be called the complete transfer relation (CTR). 3 The derivation of the CTR can be done based on the physical phenomenon that is described by the governing equations or by using the Galerkin method. The latter approach will be used herein, because of its generality. et us use the following weighting function w(x) = x φ + x φ = N φ φ. (9) Using the Galerkin projection one obtains 0 N T d φ + c 0 N T φ(x) + After integrating by parts the first term, equation (0) becomes + 0 φ () 0 φ () c x 0 x = 0 x x 0 N T d(x) = 0. (0) φ(x) = d(x). () et us now replace the function φ of the third term of the left-hand side by the expression given in equation (4), where the vector of coefficients C is written as in equation (8). Equation () becomes where c 3 G0F 8 G0F 0 8 = 36 φ () φ () G0F 8 = c 4 G0F 3 8 φ () φ () φ (3) φ (3), G0F 8 = 5 cg0f 0 8, G0F 3 8 = 50 = x x,. c G0F 8 φ () φ () d(x) () Note that the third term of left-hand side of equation () is responsible for introducing errors in FEM, because the function φ in the integral is approximated by a low degree polynomial. This 5

6 term is responsible for introducing errors in the FEM. Unlike the FEM, the AEM uses a high degree polynomial to approximate φ(x). The result of this approach is that equation () has 8 unknowns,..., φ(3). There are, however, only 4 available relations: equations () and boundary conditions (8). Consequently, 4 additional relations are necessary. Usually, when more information is needed, this information is obtained either from some relations with other elements or by accepting a reasonable approximation. Neither of these options are used in the AEM. Instead, the necessary information is obtained accurately from inside the, φ(0) element, that is, from the governing equations themselves. derivatives are φ () φ () = c d (0) d (0) Consequently, the second and third (3) and φ (3) φ (3) = c φ () φ () d () d (), (4) where d (0) := d(x = 0), d (0) := d(x = ), d () := d d(x) x=0, d () := d d(x) x=. Consequently, the equation () becomes A φ () + B φ () = C, (5) where A = c G0F c 3 G0F 8 B = 0 0 c G0F 8 + c 4 G0F 3 8 (6) C = x 0 x d(x) c 3 d (0) G0F 8 d (0) c 4 d () G0F 38 d () The unknowns φ (), φ(), φ(3) and φ (3), which we call the apparent unknowns,4 have been eliminated by using the governing equations. To illustrate the methodology, let us consider that in equation () c = 4 and d(x) = 4x 3 8x 8x 8, that is d φ + 4φ 4x3 8x 8x 8 = 0 (7) with the boundary conditions φ := φ(x = 0) = 0 (8) φ := φ(x = ) = 0. 6

7 The analytical solution of equation (7) with the specified boundary conditions is: φ(x) = (cos() 7)/ sin() sin(x) cos(x) + x 3 + x + 3x +. (9) The values of the A, B and C matrices corresponding to the seventh- and nineteen-degree interpolation function are obtained from equations (6): HIF8 A = B = C = / / HIF0 A = B = C = For comparison, let us show the A, B and C matrices obtained using a linear interpolation function (IF) A = /3 5/3 5/3 /3 B = 0 0 C = 3.6/3.8. Table shows the variation of φ () as a function of the degree of the polynomial in the HIF. This solution is based on one element. For comparison, the value of φ () calculated using the FEM with an IF is also included. The exact value is obtained from the solution (9) as φ () = The variation of the error shows that an accurate solution can be obtained with only one AEM element with HIF. φ () Error % IF HIF HIF e-3 HIF e-7 HIF e- HIF e-4 exact Table Solution based on one element. The variation of φ () as a function of the number of elements is shown in Table. Results are shown for the FEM with IF and the AEM with HIF4 and HIF8. The advantage of using the AEM 7

8 and high-degree polynomials is obvious: two AEM elements with HIF8 produce more accurate results than 00 FEM elements with IF. Four AEM elements with HIF8 produce a solution with an error of the order of 0 8 %. IF HIF4 HIF8 Number Elem. φ () Error % φ () Error % Error % e e e e- -.84e e-3 * e-4 * e-5 * e e-7 * e-3 * * * Table Variation of the value of φ () as a function of the number of elements and degree of the HIF/IF. AEM Applied to Nonlinear ODEs This section presents the methodology for solving nonlinear ODEs using the AEM. In order to solve nonlinear ODEs using the AEM, a methodology for solving linear ODEs with variable coefficients must be developed first. For this reason, the first part of this section is devoted to solving linear ODEs with variable coefficients. The second part of this section presents the methodology for solving nonlinear ODEs using the AEM. et us consider the following second-order ODE with variable coefficients a(x) d φ + b(x)dφ + c(x)φ(x) + d(x) = 0. (0) The CTR corresponding to equation (0) will be derived using the Galerkin method. reason, it is convenient to rewrite equation (0) in a transformed form For this d a(x) dφ + d ( b(x) da(x) ) ( φ + c(x) db(x) ) + d a(x) φ + d(x) = 0. () et us use for the Galerkin projection the weighting function w(x) from equation (9). The result of the Galerkin projection applied to equation () is: T d N a(x) dφ ( T d + N b(x) da(x) ) φ + T N (c(x) db(x) ) + d a(x) T φ + N d(x) = 0. () For brevity, only the derivation of the first term of equation () will be presented herein. The same methodology is applied for the derivation of the other terms in equation (). The first term of equation () will be calculated using integration by parts: T = T d N a(x) dφ d T N = a(x)dφ + T dφ N a(x). (3) The first term of the right-hand side is denoted T a and the second term is denoted T b. Taking 8

9 into account the expression of N from equation (9), the term T a becomes d T N T a = a(x)dφ = a(x) dφ. Integrating by parts again, T a becomes T a = { } da(x) φ + (a()φ a(0)φ ). Using the notations introduced in equation (5) and a (k) := dk a(0) and a (k) k := dk a(), the term T k becomes T = da(x) φ a (0) a (0) a (0) a (0) a (0) 0 φ () 0 a (0) φ (). A similar approach is used for the other terms in equation (), such that the general form of the CTR is a (0) / + b(0) a () a (0) a (0) / a(0) / b(0) a () a (0) + 0 φ () 0 a (0) + ( b(x) da(x) ) φ ( c(x) db(x) + d a(x) ) N T φ = φ () N T.(4) Having derived the CTR for linear ODEs with variable coefficients, the next step is to produce an algorithm for solving nonlinear ODEs. To present the methodology for solving nonlinear ODEs, let us choose the following second-order, nonlinear ODE φ () + (3 + 4φ () 4) + (8 40x + 85x 6x 3 + x 4 + x 5 3x 6 ) = 0 (5) with the boundary conditions := φ(x = ) = 0 and φ () := dφ (x = ) = 0. For simplicity, let us assume =. The nonlinearity is due to the 3 and 4φ () terms. In order to eliminate the nonlinearity, the terms in the parentheses (3 + 4φ () 4) will be approximated using HIF. To simplify the algebra, a third-degree polynomial HIF will be used in this example. The derivation of the vector of coefficients C is similar to the derivation presented in section, where C is given by equation (8). For a third-degree polynomial HIF, the expression of the vector of coefficients C is C = F 40 + F 4 φ () φ (), (6) where F 40 = and F 4 = 0. 9

10 Using equation (4), the HIF turns out to be the well known Hermite function φ(x) = 3 3x + x x x x x + x 3 φ () + x + x 3 φ (). (7) Replacing the terms in the parenthesis (3 + 4φ () 4) by the HIF from equation (7), equation (5) becomes where φ () + (K 0 + K x + K x + K 3 x 3 ) + (8 40x + 85x 6x 3 + x 4 + x 5 3x 6 ) = 0, (8) K 0 = 7 4, K = 4 3φ (), K = 5 + 6φ (), K 3 = 6 + 3φ (). (9) As a result, the nonlinear ordinary differential equation (5) has been replaced by a linear ODE with variable coefficients (8). The CTR of equation (8) is derived using the general form obtained in equation (4). Having b(x) = 0 and =, equation (4) becomes = 0 ( K0 + K x + K x + K 3 x 3) T N φ(x) = φ () φ () = 0 + d(x) N T. (30) After substituting the HIF, φ(x), from equation (7), the K i coefficients from equation (9) and d(x) from equation (5), the CTR equation (30) becomes ( + 0 φ () ) + /5 0 7/ /0 3/0 φ() + ( ) 3/840 φ () /0 + 49/0 /0 + 7/0 φ () 7/5 357/840 = 0, (3) 333/840 or using the notation u := and v := φ () { 336u 356uv 3v u + 48v = u + 9uv + 7v + 94u + 39v = 999 (3) Two sets of real solutions are obtained by solving the nonlinear system of equations (3) u =, v = 5 and u = 5.8, v = Only the first set of solutions leads to an acceptable solution which is φ(x) = (x 3 3x + ) + (x 3 x + x)φ () = (x 3 3x + )( ) + (x 3 x + x)(5) = x 3 4x + 5x. (33) The second set of solutions (u := = 5.8, v := φ () = 60.3) produces the solution φ(x) = 9.57(x ) (x 0.799) which does not satisfy the governing equation (5). 0

11 AEM vs. Existing ODE Solvers To compare the AEM against existing ODE solvers, let us consider the second-order ODE with variable coefficients: (+x+3x +3x 3 +x 4 ) dφ +( 5x+x +3x 3 +4x 4 ) dφ +(+x+x +3x 3 +4x 4 )φ+f(x) = 0, where f(x) has been chosen to be f(x) = x + 8x + 66x 3 59x 4 8x 5 98x 6 75x 7 44x 8 053x 9 056x 0 447x 60x 6x 3, such that the solution of the ODE is the ninth-degree polynomial φ(x) = 3 + 5x 4x + x 3 6x 4 + x 5 + x 6 + x 7 + x 8 + 4x 9. Various integration lengths have chosen with various two-point boundary conditions, as shown in Table 3. The AEM has been compared against the shooting method and the relaxation method. All the methods were implemented in FORTRAN. The shooting method used the Runge-Kutta method implemented in the odeint routine. 4 One hundred years after they have been introduced, 8 the Runge-Kutta methods are still the workhorse for solving ODEs, 4 p There were several improvements of the Runge-Kutta method, such as Runge-Kutta-Gill and (3/8) Runge-Kutta, but these modifications did not fundamentally changed the basic idea of the method. Other method for solving ODEs, such as the Bulirsch-Stoer method and the predictor-corrector method, can be faster than the Runge-Kutta in certain circumstances, but the differences are rather limited. Given these facts and due to its robustness, the fourth-order Runge-Kutta method has been selected as one of the methods to be compared against the AEM. The relaxation method used the solvde routine. 4 The computations were done on an SGI Indigo computer, with an R8000 processor. The results shown in Fig. indicate that for the same computational time, the accuracy of the AEM results with a 3th-degree polynomial (HIF4) was approximately five orders of magnitude better than the accuracy of the shooting and relaxation methods. In addition, for the same error level, the AEM with HIF4 interpolation functions produced solutions at least two orders of magnitude faster than the shooting and relaxation methods. The relaxation method diverged when the integration was equal or exceeded x =. Domain Boundary Values Case x left x right φ left φ right Table 3 Integration cases. CONCUSIONS A new method, which we call the accurate element method, has been presented herein and applied to linear and nonlinear ODEs. A key feature of the AEM was the methodology developed for eliminating unknowns inside the element by using the relations provided by the governing equations. The results of the AEM applied to a second-order ODE showed that the order of the local approximation function did not affect the number of unknowns used to discretize the differential equation. This is a breakthrough in the paradigm used to approach the numerical solution of differential equations. For a second-order ODE it was shown that the discretized solution has the same number of unknowns, whether the local approximation (or interpolation) function is a

12 0<=x<= 0<=x<= log0(error) % AEM, CF0 AEM, CF4 Relax, eps=5e 6 Relax, eps=5e 7 Relax, eps=5e 9 Shoot, eps=e, h=e 8 Shoot, eps=e, h=e 6 Shoot, eps=e 8, h=e 6 log0(error) % AEM, CF0 AEM, CF4 Shoot, eps=(e 8,e 4), h=e log0(time) sec (i) log0(time) sec (ii) 0<=x<=3 0<=x<= log0(error) % log0(error) % AEM, CF0 AEM, CF4 Shoot, eps=(e,e 8), h=e 8 0 AEM, CF0 AEM, CF4 Shoot, eps=(e,e 8), h=e log0(time) sec (iii) log0(time) sec (iv) Figure AEM vs. Relaxation and Shooting Methods

13 third-degree or a nineteen-degree polynomial. The implication of this result is that higher-degree interpolation functions can be used without increasing the number of unknowns. The AEM has been used to solve linear ODEs with constant and variable coefficients and nonlinear ODEs. For a linear ODE with variable coefficients, the AEM has been compared against the relaxation method and the shooting method. The accuracy of the AEM results with a 3thdegree polynomial (HIF4) was approximately five orders of magnitude better than the accuracy of the shooting and relaxation methods. In addition, for the same error level, the AEM with HIF4 interpolation functions produced solutions at least two orders of magnitude faster than traditional ODE solvers. REFERENCES. C. Bernardi, Y. Maday, C. Mavriplis, and A. T. Patera. The mortar element method applied to spectral discretizations. In T. J. Chung and G. R. Karr, editors, Finite Element Analysis in Fluids, The Seventh Int. Conf. on Finite Element Methods in Flow Problems. UAH Press, C. Bernardi, Y. Maday, and A. T. Patera. A new nonconforming approach to domain decomposition: The mortar element method. In H. Brezis and J.. ions, editors, Nonlinear Partial Differential Equations and Their Applications, number 99 in Pitman Research Notes in Mathematics. Wiley, M. Blumenfeld. A New Method for Accurate Solving of Ordinary Differential Equations. Editura Tehnica, Bucharest, M. Blumenfeld and P. G. A. Cizmas. The accurate element method: A new paradigm for numerical solution of ordinary differential equations. submitted for publication to the Journal of the Romanian Academy, January J. P. Boyd. Chebyshev and Fourier Spectral Methods. Springer, New York, C. Canuto, M. Y. Hussaini, A. Quarteroni, and T. A. Zang. Spectral Methods in Fluid Dynamics. Springer, New York, G. F. Carey and A. I. Pehlivanov. Computer methods in applied mechanics and engineering. Computer Methods in Applied Mechanics and Engineering, 50(-4):5 3, Dec R. Y. Chang and C. H. Hsu. A variable-order spectral element method for incompressible viscous flow simulation. International Journal for Numerical Methods in Engineering, 39(7): , Demkowicz, J. T. Oden, W. Rachowicz, and O. Hardy. Toward a universal h p adaptive finite element strategy: Part : Constrained approximation and data structure. Computer Methods in Applied Mechanics and Engineering, 77(-):79, D. Funaro, A. Quarteroni, and P. Zanoli. An iterative procedure with interface relaxation for domain decomposition methods. SIAM Journal on Numerical Analysis, 5(6):3 36, Dec S. Gill. A process for the step-by-step integration of differential equations in an automatic digital computing machine. Proc. Cambridge Philos. Soc., 95.. D. Gottlieb and S. A. Orszag. Numerical Analysis of Spectral Methods: Theory and Applications. SIAM-CMBS, Philadelphia,

14 3. A. K. Gupta. A finite element method for transition from a fine to a coarse grid. Int. J. Numer. Meth. Eng., :395 4, R. D. Henderson. Dynamic refinement algorithms for spectral element methods. Computer Methods in Applied Mechanics and Engineering, 75(3-4):395 4, R. D. Henderson and G. E. Karniadakis. Hybrid spectral element-low order methods for incompressible flows. J. Sci. Comp., 6():79, G. E. Karniadakis and S. J. Sherwin. Spectral/hp Element Methods for CFD. Oxford University Press, Oxford, K. Z. Korczak and A. T. Patera. An isotropic spectral element method for solution of the navier-stokes equations in complex geometry. Journal of Computational Physics, 6():36 38, Feb W. Kutta. Beitrag zur naherungsweisen integration totaler differentialgleichunge. Zeit. Math. Physik, 46: , C. acour and Y. Maday. Two different approaches for matching nonconforming grids: The mortar element method and the feti method. BIT, 37(3):70 738, Y. Maday and A. T. Patera. Spectral element methods for the incompressible navier-stokes equations. In A. K. Noor and J. T. Oden, editors, State of the Art Surveys in Computational Mechanics, chapter D. J. Morton, J. M. Tyler, and J. R. Dorroh. A new 3D finite element for adaptive h- refinement in -irregular meshes. International Journal for Numerical Methods in Engineering, 38(3): , Dec T. J. Oden and. Demkowicz. Advances in adaptive improvement: A survey of adaptive finite element methods in computational mechanics. In A. K. Noor and J. T. Oden, editors, State of the Art Surveys in Computational Mechanics, chapter A. T. Patera. A spectral method for fluid dynamics: aminar flow in a channel expansion. Journal of Computational Physics, 54(3): , W. H. Press, W. T. Vetterling, S. A. Teukolsky, and B. P. Flannery. Numerical Recipes in FORTRAN - The Art of Scientific Computing. Cambridge, second edition, W Rachowicz, J. T. Oden, and. Demkowicz. Toward a universal h p adaptive finite element strategy: Part iii: Design of h p meshes. Computer Methods in Applied Mechanics and Engineering, 77(-):8, J. N. Reddy. An Introduction to the Finite Element Method. McGraw-Hill, New York, second edition, E. M. Ronquist and A. T. Patera. Spectral element methods for unsteady navier-stokes equations. In Proceedings of the Seventh GAMM Conference on Numerical Methods in Fluid Mechanics, page 38. Viewig, W. Wang. Special quadratic quadrilateral finite elements for local refinement with irregular nodes. Computer Methods in Applied Mechanics and Engineering, 8(-):09 34, Feb