Spectral schemes on triangular elements by Wilhelm Heinrichs and Birgit I. Loch Abstract The Poisson problem with homogeneous Dirichlet boundary conditions is considered on a triangle. The mapping between square and triangle is realized by mapping an edge of the square onto a corner of the triangle. Then standard Chebyshev collocation techniques can be applied. Numerical experiments demonstrate the expected high spectral accuracy. Further, it is shown that nite dierence preconditioning can be successfully applied in order to construct an ecient iterative solver. Then the convectiondiusion equation is considered. Here nite dierence preconditioning with central dierences does not overcome instability. However, applying the rst order upstream scheme, we obtain a stable method. Finally, a domain decomposition technique is applied to the patching of rectangular and triangular elements. Keywords spectral, collocation, triangle, preconditioning, Poisson, convectiondiusion, domain decomposition. Introduction Pseudospectral collocation methods give good approximations to smooth solutions of elliptic partial dierential equations. However, there is a huge disadvantage as these methods are conned to rectangles. Additionally, the spectral operator is ill conditioned compared to nite dierence or nite element operators and requires preconditioning to construct an eective iterative solver. Here, we apply the standard Chebyshev collocation method for solving partial dierential equations on certain right triangles. We introduce a transformation between the triangle and the standard square where spectral collocation can be applied. This transformation maps one edge of the square onto one corner of the triangle so that the nonequally spaced collocation points cluster in that corner. In [] a dierent approach has been examined. The results are compared. This method is then applied to the Poisson equation with homogeneous Dirichlet boundary conditions on a right triangle. It is numerically shown that for smooth solutions high spectral accuracy can be achieved. Then we introduce a singularity caused by the singular behaviour of the righthand side leading to a somewhat slower convergence of the approximation. Preconditioning by nite dierences yields a condition number increasing as O(N). After that the convectiondiusion equation is considered. To overcome the instability for small we choose N to be odd (see []). Preconditioning by central nite dierences yields an unbounded condition number such that an upwind method has to be applied. Finally, domain decomposition problems are investigated. The Poisson problem is numerically solved on patchings of rectangular and triangular elements. A Dirichlet Neumann interface relaxation is iterated until continuity of normal derivatives is achieved. By numerical results the eciency of this treatment is demonstrated.
Transformation of the right triangle The standard Chebyshev collocation scheme (see []) is dened for the nonequally spaced ChebyshevGaussLobatto nodes (s i ; t j ) = (cos i N ; cos j N ) on the square [ ; ]. Using linear transforms, arbitrary rectangles can be considered. However, if we are interested in triangular domains the mapping is more complicated. In [] a mapping applying polar coordinate transformation and bending of an edge of the triangle was introduced and analyzed. Numerical results showed the eectiveness of this method. Here we consider a new transformation between the standard square R = f(x; y) j < x; y < g and the right triangle T = f(x; y) j < x; y < and x + y < g. The original mapping is given in [7] and has been changed for our purposes. The transformation reads x = 4 (x R + )( y R ); y = (y R + ) x R = x y ; y R = y and is displayed in Figure. y T x y R xr Figure : Horizontal transformation We will call this the horizontal transform as every node is actually moved horizontally. The vertical transform is x = (x R + ); y = 4 (y R + )( x R ) x R = x ; y R = y x and will be considered later. This transformation is no longer injective. We will see that this does not disturb the accuracy of our approximation. The upper edge of R is mapped onto P(,) on T. As this edge belongs to the border of our domain boundary conditions are applicable which are treated separately anyway.
Partial derivatives must be transformed, too. Using the horizontal transform we derive u x = u x = 4 y R u xr u xx = 4u x x = ( y R ) u xr x R u y = u y = x R+ y R u xr + u yr u yy = 4u y y = 4 (x R+) ( y R ) u xr x R + 8 x R+ y R u xr y R + 8 x R+ ( y R ) u xr + 4u yr y R : The Laplacian then reads as follows u = u xx + u yy = 4 4+(x R+) ( y R ) u xr x R + 8 x R+ y R u xr y R + 8 x R+ ( y R ) u xr + 4u yr y R : The Poisson problem Numerous spectral algorithms for the numerical simulation of physical phenomena demand the approximative solution of one or more Poisson problems in a bounded domain. We now study the problem u = f in T; u = on T; where T denotes the boundary of T. We apply the standard Chebyshev collocation scheme to the exact solution u(x; y) = xy(e x+y e): () This function obviously fullls the boundary condition. Table shows the discrete L error E := ku u N k N. One observes the exponential decay of the error. N E E in [] 4 :94 5 :89 4 8 :4 8:85 7 : :84 3 4:9 :78 Table : Error using horizontal transformation and [] As we see the high spectral accuracy can also be reached on the triangle T. We have the best approximation of the solution at P(,) as the collocation points cluster there. Figure shows the position of the collocation points for N= on the triangle and on the square. 3
y x y x Figure : Positions of the Chebyshev collocation nodes for N = Comparison of these results to those in [] using polar coordinate transformation (see Table ) shows that our mapping yields a faster convergence of the approximation. Here rounding error accuracy is already reached for N=. N=4 and N=8 give results which are more exact by or 5 digits. This can be explained by the position and way of numbering the collocation nodes. Figure 3 shows that the jumps occuring when changing the row (e.g. from third to fourth point) are decreasing while those in [] seem to be larger. The speed of convergence is probably inuenced by greater jumps. 7,8,9 5 4 4 5 3 3,,9 8 7 Figure 3: Order of collocation points for N = compared to [] Next we consider a singular problem where f. We compare the results for N=4, 8, and 3 to those obtained for N=3 at the xed points displayed in Figure 4. These points are the collocation nodes for N=4 which are also used for larger N divisible by 4. We expect the error to be smallest close to y= because there the collocation nodes cluster. We deal with the following nodes: P (; ); P ( p ; p p p ); P 3( ; ); P 4( p p ; ) and P 5( 4 p ; p ):
y P 3 y P 4 x P 5 P P x Figure 4: Positions of the ve nodes The approximation converges more slowly than in the last example. That makes sense because here the dierential equation and its boundary condition are not compatible any more. To get an overview we present ER = ju N u 3 j which is the absolute value of the dierence for every node, in a diagram (Figure 5).. e5 e e7 P P P3 P4 P5 ER e8 e9 e e e 5 5 5 3 N Next we choose f discontinuous: ( for y x > f(x; y) = for y x : Figure 5: Poisson problem with constant f As Figure shows the triangle is now bisected. The transformation of the line y = x on the triangle gives the hyperbola y = x+ on the square. x+3 5
y y f = f = f = x f = Figure : Transformation of the line The results can be found in Figure 7. x... P P P3 P4 P5 ER e5 e e7 e8 5 5 5 3 N Figure 7: Poisson problem with discontinuous f The approximation is relatively bad close to the separating line. Since f is discontinuous the solution of the partial dierential equation is no longer smooth and there is no high spectral accuracy any more. We only have a rst order method. Preconditioning We are interested in a good condition number of our spectral operator which does not increase too fast such that ecient iterative solvers can be found. Here the maximum eigenvalues of the spectral Laplacian on the triangle scale as O(N 8 ) (Table ). On the square one has O(N 4 ) which is certainly preferrable. We are looking for a preconditioner to improve the condition so that it scales as O(N) or even independently of N. A good preconditioner also has to be a good approximation of the inverse of the spectral operator. We found that the condition number is already
reduced if we multiply the operator by ( y R ). The partial derivatives contain this factor in the denominator. For y close to the inuence of the appropriate partial derivative is extremely high. The discretized operator is called L ;SP. Table 3 shows max := maxfjj j eigenvalueg, min := minfjj j eigenvalueg and cond max min. Here the condition number scales as O(N 4 ). N max min cond max =N 8 4 5:39 3 5:3 : :8 8 : 4:94 :3 4 :7 :7 8 4:93 5:49 : 3 :8 4:93 :39 9 : Table : The spectral operator L SP N max min cond max min cond 4 :3 5:4 :8 :5 5: : 8 9:77 3 5:35 :83 :89 3 4:58 4:5 :5 5 5:3 :94 3 3:7 4 4:39 7: 3 3 :5 5:3 4:7 4 4:93 5 4:9 :5 5 Table 3: The spectral operator L ;SP and results in [] Our results are comparable to those in []. We now study the nite dierence preconditioner L F D which is the discretization of the Laplacian by second order nite dierences. The rst and second derivatives are where w (s j ) = ( j w(s j ) ( j j )w(s j ) + j w(s j+ )); w (s j ) = j ( j w(s j ) ( j + j )w(s j ) + j w(s j+ )) j = j = ; s j+ s j for j = ; : : : ; N (see []). s j+ s j Table 4 shows the improved results. N max min cond max min cond 4 :73 : :73 :7 :99 :73 8 :3 :89 :4 : :99 :3 :5 :7 3:53 :4 :8 3: 3 :9 : 4:89 :83 : 4:3 Table 4: (L F D ) L SP and results in [] 7
Now we obtained a condition number scaling as O(N). We could construct an eective iterative solver now. Figure 8 shows the positions of the eigenvalues for N=3. Their imaginary parts are fairly small and the real parts are contained in [:5; 3]...4. IM..4..5.5.5 3 RE Figure 8: Eigenvalues of (L F D ) L SP for N = 3 One could apply higher order FDmethods for an even better condition number. However, this would result in an extended eort for solving the FD problem. In summing up, we state that this transformation between triangle and square gives comparable or better results than the transformation by polar coordinates in []. 8
The convectiondiusion equation Modelling of purely convectional or convection dominated processes is a central problem in areas like e.g. meteorology or investigation of aerodynamical or geophysical ows. A model boundary value problem is the convectiondiusion equation u + au x + bu y = f in T; u = on T; which can be used for describing the expansion of temperature in a uent. Temperature expands uniformly diusive in every direction which is expressed by u. It is spread by current, too, called convection and is described by au x + bu y (a and b being the velocities in x and in y direction). As usual, is the viscosity of our material and represents a measure for interior friction. As the partial dierential equation is of dierent type for > and = (in the rst case it is elliptic and in the latter it is hyperbolic) we talk about singular behaviour. In the interior of our domain u and u are close together but getting to the boundary they dier extremely. Homogeneous Dirichlet boundary conditions are not applicable to hyperbolic problems such that we have to deal with boundary layers now. Boundary layers are environments where derivatives of u scale as O( ). Those systems are also called sti systems. Unphysical oscillations occur in the numerical solution and the discretization is instable. Figure 9 shows the situation in D (see [4]). u(x) numerical solution exact solution x Figure 9: Boundary layer We are looking for a method to resolve the boundary layers. There are schemes which still use spectral methods like adding articial viscosity, spectral viscosity or streamline diusion. However, here we only choose odd N. Oscillation always arises and is increased for even N with N while this is not the case if N is odd. The following table contains the discrete L error for decreasing which develops when discretizing the convectiondiusion equation by spectral collocation. Here we choose (a,b)=(,) 9
and (,) as these two cases are good representatives for other choices of (a,b). We have tested the algorithm with example (). In the case of pure convection ( = ) the method is unstable. With decreasing the singular behaviour is increasing and one has to choose a ner grid (larger N) to obtain results comparable to =. As mentioned above, we now choose N odd which usually leads to a decreased error. This behaviour was analyzed in [] in D on the square. It can be transferred to the triangle with only few restrictions concerning the choice of parameters. If N is even there exists an interpolation polynomial which fullls the boundary conditions and whose derivative vanishes at the collocation points. This polynomial is responsible for the instability. On the contrary, if N is odd one nds the proof in [] that this polynomial does not exist. Apparently, there are parameters (a,b) for which the spectral method is unstable even for odd N. For the stable case (,) we actually have the regular operator x on the square multiplied with a factor. For (,) we have exactly that combination of the rst derivatives on the square where there are at least two equal rows in the derivative matrix. The partial derivatives are based on the matrix D N. As the collocation points on the square are symmetric (for every positive node we nd a corresponding negative one) there is annulation in the derivative matrix. The following example for N=3 shows the connection. u y = x R+ u y xr R + u yr yields the derivative matrix B s ( s ) s (s +) ( s )(s s ) (s +) ( s ) ( s )(s s ) s ( s )( s s ) s s s s s s s s ( s )( s ) s s s s (s +) ( s )(s s ) (s +) ( s )(s s ) s ( s s ) s As s = s (symmetry) we have equal second and third row and the matrix is singular. (,) shows the same behaviour. For (,) we do not have annulations and the method is stable. Table 5 displays the results. (,) can be stabilized by using the vertical transformation where x and y are exchanged. E for (a; b) N = = = 4 = = (; ) 3 :75 4 : 3 :5 3 :5 3 :5 3 7 8:75 4:8 9 7:77 9 7:77 9 7:77 9 5 3:77 5:3 7 :8 :5 :4 3 5:8 :7 :7 5:4 3:7 ( ; ) 3 :5 4 3:75 3 :9 :8 :9 3 7 8:7 5: 9 :8 7 :8 5 :59 8 5 : 7:53 7 :8 :7 4 5:75 3 4:5 :7 4:74 :9 4 4:3 Table 5: Error for the convectiondiusion equation Next a constant righthand side is considered. Dierential equation and boundary condition are not compatible here, i.e. u + au x + bu y = in T; C A
u = on T: Table shows the dierence ER between u 3 and u N at P(,). P(,) is in the center of the triangle and therefore far away from any boundary. It is the only collocation point (out of PP5) where stability is achieved for (,) for small. ER for (a; b) N = = = 4 = = (; ) 4 :34 4 :99 9:53 :7 4: 8 :8 5:99 7:8 4:83 3 :4 : 8 :4 3 : 7:3 3 8: 3 8:9 :8 7 : :58 3 :89 ( ; ) 4 :8 5 : :34 :35 3 :79 5 8 :5 :7 :43 :4 :7 5 : 8 : 3 :75 3 :45 :7 5 3 8:88 :3 5 : :7 : 5 Table : Error for constant f in P A discontinuous righthand side f(x; y) = ( for y x > for y x yields an even slower convergence rate than the last example (Table 7). ER for (a; b) N = = = 4 = = (; ) 4 :53 3 :8 :8 : 3:9 8 4:9 4 : 7:4 : 8:5 :3 4 3:4 4 4:4 :37 8:89 3 9:35 5 5:88 4 :89 :73 3 3:8 ( ; ) 4 :7 3 :39 5:4 5:48 3:43 4 8 4:94 4 :8 :93 :9 9:3 4 :53 4 4:5 3 :35 3: 5:3 4 3 : 4 :79 3 3:4 3:8 8:9 4 Table 7: Error for discontinuous f in P Preconditioning For the construction of an eective iterative solver we now examine the condition number of the spectral operator L ; of ( y R ) ( + au x + bu y ).
3 eps= 3 4 8 4 Figure : Eigenvalues of L ;SP for N=5, (a,b)=(,) Figures and show the positions of the eigenvalues for = and = for (a,b)=(,), N=5. = = (a; b) N max min cond max min cond (; ) 3 : 5:9 3:7 8:7 :43 3: 7 5:7 3 5:3 : 8:3 7: :3 5 : 5 5:3 : 3 4:43 :73 :57 3 3 : 5:3 4:5 4 :9 3 4:4 4: 4 ( ; ) 3 : 5:8 3:8 3:7 : 7 5:75 3 5:3 :7 5:3 : :93 7 5 : 5 5:3 :7 3 :8 : 4:33 7 3 : 5:3 4:5 4 :9 3 :5 :98 8 Table 8: L ;SP Table 8 gives max ; min and cond and demonstrates that there really is an eigenvalue close to for (,). Applying the inverse of the FD operator L F D as preconditioner, we observe decreased condition number if = while for small, max is unbounded for (,). This preconditioner obviously does not stabilize. Figures and 3 show the positions of the eigenvalues. For small they are relatively dense positioned with few peak values. In general, FD methods applied to singular disturbance problems are stable if the step size h i <. Contrary, if h i they are unstable. To obtain stability one could increase the number of collocation points which reduces the step size. A more promising attempt is the use
5 4 eps=^ 3 3 4 5 4 4 8 Figure : Eigenvalues of L ;SP for N=5, (a,b)=(,) of the upwind method. The rst derivatives x and y, the convectional part, is discretized by onesided streamdirected nite dierences while the diusive part is treated with central dierences. We lose one order in convergence but stability is achieved. We have a u x = a 4 y R u xr and b u y = b ( x R + y R u xr + u yr ): According to the factor the derivatives u xr and u yr are discretized by left or rightdierences in stream direction: u xr (x i ; y j ) = 8 < : u(x i+ ;y j ) u(x i ;y j ) x i+ x i if a u(x i ;y j ) u(x i ;y j ) x i x i if a < for i = ; : : : ; N or i = ; : : : ; N. Analogously for u yr. The upwind method is not uniformly convergent. An adaptive renement might help here. Figures 4 and 5 show that by applying the upstream method the positions of the eigenvalues have completely changed for small. They are complex, bounded and symmetric. Table 9 gives the numerical results for the upstream scheme. 3
.5.4.3.....3.4 eps=.5..8..4..8..4. Figure : Eigenvalues of L F D L SP for N=5, (a,b)=(,) 4 3 eps=^ 3 4 3 4 5 7 8 9 Figure 3: Eigenvalues of L F D L SP for N=5, (a,b)=(,) 4
.5 eps= eps=.5.5.5.5.5.5 Figure 4: Eigenvalues of the upstream operator for N=5, (a,b)=(,).5 eps= eps=.5.5.5.5.5.5.5 Figure 5: Eigenvalues of the upstream operator for N=5, (a,b)=(,) 5
(a; b) (; ) ( ; ) N max min cond max min cond 3 :4 9:8 :5 :4 9:35 :5 7 : 8: :39 : 8:58 :4 5 :39 :99 3:4 :39 :97 3:43 3 :85 5:9 4:8 :85 5:9 4:8 3 :4 3:4 :98 :8 : :58 7 :37 :99 4:5 :4 :7 5:49 5 :8 4:9 5:34 :9 4:8 4:55 3 :37 4:34 5:47 :37 5:8 4:7 4 3 : 3:5 : 3:33 :4 3 :35 7 :4 :5 7: :9 3: 3 3:84 5 :9 8:44 5:3 :3 8:37 3 :5 3 :8 8:8 :5 :8 :74 :5 3 :7 3:5 : 3:33 :4 5 :34 4 7 :5 :5 7:5 : 3: 5 3:85 4 5 :3 7:79 :8 :3 8:4 5 :55 4 3 :4 5:7 4:7 :34 :79 4 7:5 3 3 :7 3:5 : 3:33 :5 7 :48 7 :5 :5 7:5 : :84 8 :48 7 5 :3 7:78 :8 :3 : 8 :98 7 3 :4 5:9 4:8 :34 5: 7 :58 Table 9: Upstream method It is not satisfactory that there are cases (eg. (,)) in which no stability can be achieved. A possibility to overcome this may lie in the introduction of an additional collocation point. The system is then overdetermined. This method has been examined and successfully applied on the square in []. A further method may be the use of staggered grids which possibly leads to min >. Two dierent sets of grids are used one for the solution and the other one for its derivative. For the advectiondiusion equation there were positive results in []. Instead of using the Gauss algorithm for solving the linear systems, one could apply iterative methods. As many other iterative methods do not support complex eigenvalues we recommend the use of the GMRES method (see [5]) a method of minimized residuals. The linear system Bv = g where B is a nonsymmetric and large matrix is solved as follows. v is the initial solution, r = g Bv and we dene the mth Krylov space K m := spanfr ; Br ; : : : ; B m r g. Then we nd the approximation v m v + K m such that the mth residual r m fullls jr m j = min!. Domain decomposition We are now interested in applying the spectral method to more complex domains. We use the patching method (see [3]) where the domain is separated into square or triangular subdomains on which GaussLobatto nodes are dened. The dierential equation is solved at the interior nodes. At the interface we require continuity of the solution and its normal derivative. We
consider the Poisson equation with Dirichlet boundary condition u = f in ; u = g on : At the interface between two subdomains, information is exchanged until continuity is reached. In one direction Dirichlet information is transfered and in the other direction it is Neumann information. We use an interface relaxation as proposed in [3] i.e. at the Dirichlet side we hand over a weighted sum of subsolutions at the interface. We iterate until the error at the interface is smaller than 4. Thus we iteratively solve a sequence of Dirichlet Neumann problems. We begin with a domain composed of one patched triangle and square = T [ R while T = f(x; y) j < x; y < and x + y < g and R = f(x; y) j < x < and < y < g: y T R x Figure : Domain The interface is = (; ) fg (Figure ). Initial conditions are u = u u = g on. We then iterate on and and u m = f in T; u m = g on T n ; u m = m u m + ( m )u m on u m = f in R; u m = g on R n ; y um = y um on : Here m denotes the relaxation parameter which is chosen dynamically. This dynamical choice usually accelerates the convergence. m = is the unique number which minimizes k z m () z m () k where z m () = u m + ( )u m. m is calculated by m = (em ; em em ) k e m ; em k 7
where (:; :) denotes the discrete L inner product and e m i = u m i u m i for i = ; is the dierence of two consecutive iterates on the two subdomains. m should be in (; ]. We cannot use example () because this function vanishes at the interface. Therefore no new information is exchanged which makes an iterative method superuous, as it converges after the rst step. Thus we introduce the following oscillating example u(x; y) = sin(x) sin(y + ): () 4 N It E T E R 4 5 :3 :8 8 7 :4 5 :4 5 7 8: 3 4:4 3 3 7 : 4 : 4 Table : with () Table shows the number of iterations and the discrete L error on square and triangle. We reach the tolerance after relatively few steps. The convergence is fairly slow because of the oscillatory behaviour of the solution. The number of iterations is constant and independent of N. Machine accuracy is reached for N=. The second domain to be studied consists of and an additional triangle T attached to the already existing one (Figure 7). y T T R x Figure 7: Domain We begin with triangle T with interface boundaries = (; ) fg and = fg (; ) and = [. Then we solve on R and T. This should be realized on a parallel computer. The algorithm reads u m = f in T; 8
u m = g on T n ; u m = m u m + ( m )u m on ; u m = m u m 3 + ( m )u m on ; and and u m = f in R; u m = g on R n ; y um = y um on ; u m 3 = f in T ; u m 3 = g on T n ; x um 3 = x um on : N It E T E R E T 4 5 :4 9: 3 :5 8 83 :98 5:7 :8 5 84 :85 3 4: 3 : 3 9 :3 3 8:3 4 9:9 4 Table : with (3) Initial values are analogous to the last example. We apply this algorithm to the example u(x; y) = sin(x + 4 ) sin(y + ): (3) 4 The results are listed in Table. The number of iterations is extremely increased if a further triangle is added. Unfortunately, i m tends to leave the interval (; ]. Whenever this happens, the following approximation is worse than the one before. Nevertheless, the method nally converges. This dynamical choice of i m is not optimal. We have derived results for xed i m = in Table. N It E T E R E T 4 34 4:89 4 :8 4 :5 3 8 4 9:9 9 :3 9 :8 8 4 :78 3 3:7 4 :83 3 3 87 9:8 3 8:4 4 : Table : with (3) and m = :5 The number of iterations is smaller and there are no 'backward steps' any more. 9
Finally, we study the domain = R [ T [ T [ T 3 [ T 4 (T i triangles) which is symmetric to the origin (Figure 8). We consider the following example u(x; y) = sin(3x + 4 ) sin(3y + ): (4) 4 y x Figure 8: 'wind wheel' The algorithm is analogous to the last one and we rst solve on the square and then on the triangles. The results in Table 3 are fairly good for symmetry reasons considered that we now deal with ve subdomains. The number of iterations is constant and machine accuracy is reached for N=. N It E R E T E T E T 3 E T 4 8 75 9: 5:8 7:55 4:89 7:3 7 4:33 5 :8 5 4:7 5 :79 5 4:3 5 3 9 3:83 3 5: 4 : 3 : 4 8: 4 Table 3: with (4) Summing up we constate that this spectral method is eective for domain decomposition problems, too. Now, we can also deal with partial dierential equations on complex domains using spectral methods as long as those domains can be separated into rectangular and triangular elements.
Bibliography [] H. Eisen, W. Heinrichs, A new method of stabilization for singular perturbation problems with spectral methods. SIAM J. Numer. Anal. 9, pp. 7 (99). [] D. Funaro, A fast solver for elliptic boundaryvalue problems in the square. Comput. Methods Appl. Engrg., pp. 5355 (994). [3] D. Funaro, A. Quarteroni, P. Zanolli, An iterative procedure with interface relaxation for domain decomposition methods, SIAM J. Num. Anal., 5, pp. 33 (988) [4] M. Griebel, T. Dornseifer, T. Neunhoeffer Numerische Simulation in der Stromungsmechanik. Vieweg Lehrbuch, 995. [5] W. Heinrichs, Defect correction for convectiondominated ow. SIAM J. Sci. Comput., 7, No. 5, pp. 89, 99. [] W. Heinrichs, Spectral collocation on triangular elements, J. Comp. Ph., 45, pp. 743757 (998) [7] S. Sherwin, G. Karniadakis, Triangular and tetrahedral spectral elements. ICOSA HOM'95: Proceedings of the third international conference on spectral and high order methods, 99 Houston Journal of Mathematics.