MSNFD : A Higher Order Finite Difference Method for Solving Elliptic PDEs on scattered points (An Extended version of our paper submitted to ICCS 2011) January 21, 2011 S. Chandrasekaran, K. R. Jayaraman, M. Gu, H. N. Mhaskar and J. Moffitt Abstract This paper presents a method of producing higher order discretiza- tion weights for linear differential and integral operators using the Min- imum Sobolev Norm idea[1][2] in arbitraty geometry and grid configura- tions. The weight computation involves solving a severely ill-conditioned weighted least-squares system. A method of solving this system to very high-accuracy is also presented, based on the theory of Vavasis et al [3]. An end-to-end planar partial differential equation solver is developed based on the described method and results are presented. Results presented in- clude the solution error, discretization error, condition number and time taken to solve several classes of equations on various geometries. These results are then compared with those obtained using the Matlab’s FEM based PDE Solver as well as Dealii[4]. 1 Keywords PDE Solver, Numerical Discretization , Sobolev Norm, Higher Order, Finite Difference , weighted least squares , parallelism , Python 0 The authors acknowledge and thank the National Science Foundation for its grant no 0830604, The NSF Teragrid project and The SDSC Triton Supercomputing resource. Also, many thanks to Stefan Borieu for his support in providing access to the above supercomputing resources. The research of H. N. Mhaskar was supported, in part, by grant DMS-0908037 from the National Science Foundation and grant W911NF-09-1-0465 from the U.S. Army Research Office. 1
Transcript
MSNFD : A Higher Order Finite Difference
Method for Solving Elliptic PDEs on scattered
points (An Extended version of our paper
submitted to ICCS 2011)
January 21, 2011
S. Chandrasekaran, K. R. Jayaraman, M. Gu, H. N. Mhaskar and J. Moffitt
Abstract
This paper presents a method of producing higher order discretiza-tion weights for linear differential and integral operators using the Min-imum Sobolev Norm idea[1][2] in arbitraty geometry and grid configura-tions. The weight computation involves solving a severely ill-conditionedweighted least-squares system. A method of solving this system to veryhigh-accuracy is also presented, based on the theory of Vavasis et al [3]. Anend-to-end planar partial differential equation solver is developed basedon the described method and results are presented. Results presented in-clude the solution error, discretization error, condition number and timetaken to solve several classes of equations on various geometries. Theseresults are then compared with those obtained using the Matlab’s FEMbased PDE Solver as well as Dealii[4].
0The authors acknowledge and thank the National Science Foundation for its grant no0830604, The NSF Teragrid project and The SDSC Triton Supercomputing resource. Also,many thanks to Stefan Borieu for his support in providing access to the above supercomputingresources. The research of H. N. Mhaskar was supported, in part, by grant DMS-0908037 fromthe National Science Foundation and grant W911NF-09-1-0465 from the U.S. Army ResearchOffice.
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2 Introduction
This paper introduces a higher order finite difference scheme for solving two di-mensional elliptic PDEs based on Minimum Sobolev Norm (MSN) interpolation[1][2].The MSN scheme is a higher order interpolation idea, that suppresses RungeOscillations by minimizing an appropriately set up Sobolev Norm of the inter-polant. We extend the idea to setup a weighted least squares problem for weightsthat approximate linear differential and integral operators. The ill-conditionedsystem that arises in the process is solved using a Complete Orthogonal Decom-position method. We present a simple, yet effective orthogonal decompositionmethod that we call CODA. With a method to approximate differential oper-ators in hand, we present a local Finite Difference like method to solve planarelliptic PDEs. The method described in this paper is fairly generic, basis inde-pendent and extends to variable coefficient, higher order and higher dimensionalproblems easily.
The underlying solution is assumed to be discretized at prescribed gridpoints. At each interior grid point, we consider a local neighborhood of gridpoints. We then use sample values at these points to approximate the underly-ing differential operator using Minimum Sobolev Norm weights. We believe thatthe order of the method depends on the number of neighbors chosen. A sparsebanded matrix corresponding to the discretization of the PDE is assembled, andsolved. Appropriate balancing is applied to control the condition number of theresulting matrix. The obvious motivation for a higher order method is that itneeds a coarser grid to solve the PDE at hand to a given accuracy than a lowerorder method. We believe that the fineness of discretization required beyondthe Nyquist rate to resolve the solution underlying a PDE to a given accuracyreduces as the order of the method increases.
The question of whether a higher order method is faster to get to a givenaccuracy remains though, as a higher order method may lead to a significantincrease in computational complexity. The simplest manifestation of this couldbe the fact that we have a denser system to solve, since we use more weights toapproximate an operator in the vicinity of a point. We assume that the cost ofweight computation is amortized by the sparse system solve cost. For regulargrids, one could exploit the regularity of the stencils to avoid computations. Alsothe weight computations are easily parallelizable even in absence of regularityand are reusable for a given geometry/grid. Let the final sparse system to besolved be
AN×Nx = b,
where A is the sparse matrix discretizing the PDE, x is the solution vector, and bis the appropriate right hand side. If we use L2 weights per row at maximum toapproximate some operator through A then A is still sufficiently sparse as longas L2 N . Assuming that we consider appropriate reordering for optimality,we believe that the time taken to solve the system is O(L3N1.5). In the maximalcase, we can have an N point neighborhood, and so each equation would haveN weights, giving us the dense compute time O(N3).
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We hypothesize that the order of the method is given by k = L+12 for a
regular grid using an L point, centered, square neighborhood. Since the numer-ical instability in computing the weights increases with L, we believe that thereis a maximum order beyond which the numerical accuracy begins to diminish.Also, considering the asympotic cost of solving the final system, we see that,the system becomes more and more dense, and a sparse solver continues to loseits efficiency. Under our assumption above, we can see that the flops requiredto solve an N ×N sparse system arising out of a kth order method would thenbe O((2k − 1)3N1.5). The accuracy achieved with a kth order method with anN0.5 ×N0.5 grid is O(N−k/2). Therefore, to get to an accuracy of εreq,
εreq = N−k/2 ⇒ N =(
1εreq
) 2k
.
The flops needed with a kth order method to get to εreq digits of accuracy wouldthen be
O((2k − 1)3(εreq)−3/k).
From Figure 2, it may be inferred that the computational advantage of thehigher order method diminishes beyond an order of about 10.
0 10 20 30 40 50100
1010
1020
1030
1040
1050
Order
Flop
s
Flops Vs Order
!req=1e-2!req=1e-4!req=1e-8!req=1e-12!req=1e-16
Figure 1: FLOPS vs Order of method for εreq = 10−16
Section 3, briefly introduces the MSN interpolation idea, and sets up theweighted LS problem corresponding to the approximation of linear operators.It then discusses the CODA method to solve the ill-conditioned WLS problemaccurately using recursive singular value decompositions. Results indicating thesuccess of CODA are presented. Section 5 discusses the discretization methodto setup a local Finite Difference like system, and the related issues. Section7 presents a host of results obtained using the PDE solver over various prob-lems and geometries. Each of these problems and geometries are solved usingvarying neighborhood sizes corresponding to varying orders. The associated
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sparse matrix bandwidth and sparse solve time are also discussed. Section 8provides conclusions and possible extensions. The Python and Matlab basedPDE Solvers will be made available through our group website [5]. The proofthat the local discretization of the PDE converges is presented in Section 6.
Related closely to our work is the work of Wright and Fornberg [6] whichdescribes a compact finite difference formulation using radial basis functions.[6] pays particular attention to the basis function, and interest is exhibited inretrieving standard FD weights. Also, [6] uses Hermite interpolation ideas incomputing the weights to improve the discretization accuracy. Furthermore,their method is a “mesh free” method, where in scattered data stencils in thelocality of collocation points are selected for accuracy. Our method is basisindependent, and provides for the possibility of adaptive gridding and selectiveneighborhood selection as well. For convenience we use a square window andregular grids on arbitrary geometry to solve PDEs. Comparable Higer Ordermethods in the FEM paradigm are the Deal II [4] and the Hermes [7] projects.
3 MSN Discretization
The MSN interpolation scheme computes an interpolating polynomial of orderhigher than the number of interpolating nodes, whose sobolev norm is minimum[1][2]. Traditional lagrange interpolation uses a polynomial of order equal tothe number of interpolating nodes. It is known that, except under particularcircumstances, such interpolation schemes produce divergent interpolants [8][9].The MSN scheme produces a polynomial of order much larger than the numberof interpolating points. Of course, there are infinitely many such polynomialssince the linear system set up to solve for the interpolant would be under-determined. We use the additional degrees of freedom to pick that interpolantwhose sobolev norm is minimum. Let V refer to a Vandermonde matrix with aChebyshev basis, f the N vector corresponding to N equi-spaced sample values,a the M N vector of spectral coefficients, s the parameter corresponding tothe Sobolev space containing the interpolant, and Ds an appropriate diagonalmatrix. M depends on the minimum spacing between the samples and thenumber of samples. Then, MSN interpolation corresponds to the optimizationproblem
arg minV a=f
‖a‖2s = ‖Dsa‖22 =M−1∑m=0
(‖m‖22 + 1)s/2|a|2. (1)
Here m is a multi-index depending on the dimensionality of the problem. Intu-itively speaking, MSN finds an interpolant that has minimum energy in higherfrequencies. This in turn corresponds to the smoothest interpolant in the senseof the above sobolev norm. It has been shown that there exists a unique solu-tion to (1) [1]. It is also shown that the interpolant converges point-wise to theunderlying function in the vicinity of the samples, provided the right Sobolevspace is chosen. Note that this convergence result is independent of the under-lying sample distribution and illustrates the local convergence property of MSN
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interpolants.
4 Stabilizing MSN Computation:CODA
While (1) is setup and solved as an under-constrained problem for which wefind the minimum norm solution, a corresponding dual Weighted Least Squares(WLS) problem can be specified as follows. We compute the MSN weights wl
such thatwl = arg min
w‖D−1
s (V TN×Π(M)w − V T
1×Π(M)(xl))‖2,where V is the Chebyshev Vandermonde matrix at (xi)N
i=1, an N point neigh-borhood of xl. Note that
f(xl) =N∑
i=1
wl(i)f(xi).
The MSN interpolation problem becomes ill-conditioned with increasingproblem size N and the Sobolev parameter s. For a local interpolation, Ndoes not need to be too large; as mentioned in the introduction, there is no sig-nificant computational advantage beyond a 10th order method. As per [3], for adiagonal ill-conditioned positive matrix scaling an otherwise well conditioned LSsystem, the solution can be computed stably. We present a poor man’s completeorthogonal decomposition based on SVDs to solve such a system. Consider anygeneral weighted LS problem of the form
arg minx‖W (Ax− b)‖.
Let y be the computed solution, and let y be the true solution. Then, the goalis an algorithm that achieves a floating point accuracy bounded independent ofW as below.
‖y − y‖ ≤ εmachf(A)‖b‖,where f(A) is a function of A independent of W . Unpivoted QR factoriza-tion of such ill-conditioned WLS systems are seen to suffer from numericalinaccuracies[3]. Hough and Vavasis proposed the complete orthogonal decompo-sition idea which uses a QR factorization with column pivoting which convertsthe given LS problem to one that is well conditioned up to a column scaling.Traditional algorithms do well to solve such systems to good floating point ac-curacies, and so an error bound of the form above is achieved. The key ideahere is to change the ill-conditioned row-scaling into an ill-conditioned columnscaling. Instead of a QR factorization with column pivoting as proposed byHough and Vavasis, we use the SVD. The theory by Vavasis et al. also seeksthat the weights be ordered; this is taken care by an apriori permutation.
We assume in this description that the factor W has been multiplied intothe equation. Consider a singular value decomposition
A = UΣV H .
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Then,AV = UΣ.
If we now solve for the modified LS problem
UΣx =LS b,
where x = V Hx, then a traditional LS algorithm works well under floating pointerrors as well. However, the SVD algorithms achieve only backward error sta-bility. Hence, we use a recursive SVD refinement approach in order to producea numerically accurate complete orthogonal decomposition. This SVD basedtechnique is described below. We require a numerical threshold η = O(1) thatspecifies the refinement. We assume that A has full column rank and well con-ditioned. This assumption is indeed valid for the MSN method assuming theunderlying grid points that produce the Chebyshev Vandermonde matrix arethemselves well spaced.
We first compute the SVD
A = UΣV H
and apply V to A, to get A = AV . Let Ao, Vo be empty matrices to accumulateresults.Also, let
k = arg minlσl <
σ0
η.
If such a k exists, then we split A as
A = [ A1 A2 ]
where A1 is the first k − 1 columns of A. We also split V as
V = [ V1 V2 ]
where V1 has the first k − 1 columns of V . We then refine A2 by computing itsSVD again as
A2 = U ΣV H
and try to split A2V , V2V as above. We accumulate Ao = [ Ao A1 ]. Wealso accumulate Vo = [ Vo V1 ]. The iteration stops when no such k existsas described above. At the end of the iterations, we compute the orthogonaldecomposition Ao = QR. Now Q,R, Vo specify a complete orthogonal decom-position.
In order to test the above algorithm we assume a known random solution tothe 2D MSN system of size 784×49 and solve this system using the factorizationdeveloped above. The threshold η used for CODA refinement was 10. Thealgorithm was observed to be sensitive to the ordering of the weights in theill-condition system. To this effect, permutations corresponding to ascendingand descending order of weights in D were employed to test the solver. Figure2 summarizes the results obtained with various orderings. In Figure 2, we
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compare three pairs of results. A QR based least squares solver in MATLABis taken as the reference for comparison. The first pair compares unorderedweights between QR and CODA. The second pair compares them with theweights sorted in ascending order, while the third pair compares them withthe weights in descending order. It is clearly evident that the CODA algorithmwith descending order of weights outperforms all other methods considered. Ourexperiments seem to indicate that descending order works best.
0 10 20 30 40 50 6010 15
10 10
10 5
100
105
1010
1015
1020Comparison of Max relative error in solving 2D MSN LS System
Figure 2: Results of CODA and variants to solve the 2D MSN system
Figure 3: The stencils used for discretization in the vicinity of points
5 Discretization of two dimensional Elliptic PDEsusing the MSN Scheme
Consider a general 2nd order elliptic PDE, given by:
∇.A∇u+ bT∇u+ cu = f (2)
We wish to solve (2) in a compact region Ω ⊂ R2 subject to the boundarycondition
eT∇u+ du = g (3)
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on the boundary ∂Ω. We consider a discretization of (2) at NI points xik ∈
Ωo, k = 1, 2, ...NI . We call these the interior points. We also consider a dis-cretization of (3) on ∂Ω at NB boundary points, xb
k, k = 1, 2, ...NB . We wishto solve (2) for u(xi
k) and perhaps u(xbk) depending on the boundary conditions.
At each xik, we consider a neighborhood of size corresponding to some order
d of local approximation. For a square stencil and equi-sampling, we consider asquare of side (2d − 1)h, where h is the sampling distance. Figure (2) depictsthese neighborhood stencils at a far interior stencil (not involving any boundarypoint) as well as in the vicinity of the boundary.
We generalize the PDE problem and construct local approximation weightsin a manner below. Let Ω ⊂ R2 be a compact region, L be a differential operatoron the interior of Ω , B be another suitable boundary operator. Then we wishto solve the boundary value problem,
Lu = f : Ωo, Bu = g : ∂Ω (4)
Let C be a finite collection of “grid points” in Ω, δ > 0. Let
B(x, δ) = y ∈ R2 : ‖y − x‖∞ ≤ δ
We fix some x∗ , and consider the set
C ′(x∗) =‖x− x∗‖∞
δ: x ∈ C ∩B(x∗, δ)
Let φk be an orthonormal set on [−1, 1]2, M = M(x∗) be the cardi-
nality of φk, where M is computed to be the mesh norm[1] of C ′(x∗) =y0,y1, ...yN−1. Let N be the cardinality of C ′(x∗), M > N .
We find weights such that
w(x∗) = arg minw∈RN
M∑l=1
(1 + ‖l‖22)−s/2
(N−1∑k=0
wkφl(yk)− Lφl(0)
)2 (5)
In order to discretize (2) and (3) at a point x∗, we place a stencil B(x∗, (2d−1)h), and at C ′(x∗) we evaluate the Chebyshev Vandermonde Matrix V . Theoperator L is applied to V (0) since x∗, the stencil center is mapped to the origin.Then, the above minimization problem problem becomes
arg minw‖Ds
(V Tw − L(V T (0))
) ‖2where Ds is the diagonal matrix with weights (1 + ‖l‖22)−s/2. As described inthe previous section, we solve this equation using the CODA method. Herethe solution w specifies the linear combination of u(C ′(x∗)) that approximatesL(u)(x∗).
Let N = NI + NB . Let F2 be the NI ×N sparse matrix that contains theweights corresponding to ∇.A∇u. The lth row of F2 contains the weights that
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approximate ∇.A∇u(xil) using u(C ′(xi
l)). Let Fxi , Fyi correspond to the NI×Nmatrices that contain the weights for ∂
where Fxb , Fyb are NB ×N sparse matrices that represent the operators ∂∂x ,
∂∂y ,
at the boundary points. Let Ci, Db represent appropriately sized diagonal ma-trices representing coefficients c, d in (2) and (3) respectively. The grand systemthat we solve is now setup as follows.
Fu =(F2 +B1Fxi +B2Fyi + Ci
E1Fxb + E2Fyb +Db
)u =
(fNI
gNB
)(8)
To reduce the condition number of F , we perform a row 2-norm scaling. Fullscale equilibration experiments are yet to be explored since the simple scalingseems satisfactory in controlling the condition number. Let this diagonal scalingbe represented by D1. Then we have a system
D1Fu = D1
(fNI
gNB
)(9)
Since fNI and gNB are at scales O(h2) and O(1) respectively, we setup andsolve for two separate systems of the form:
D1F[u1 u2
]= D1
[fNI 00 gNB
](10)
The overall solution is then reconstructed as u = u1 + u2. If the conditionnumber induced by the operator is high, as with the Bi-Harmonic operator, orsingularly perturbed operators, we need to resort to iterative refinement. Theproof that the local discretization of the PDE converges with a high order followsfrom the fact that the local interpolant converges with a high order as well andis presented in [10].
6 Local Convergence
In this section, we present an extension to Theorem 2.2b in [1] that describeslocal convergence of the MSN interpolant under linear differential operators.
Let Πm be the class of all (univariate) algebraic polynomials of degree atmost m, and for a < b, f ∈ C[a, b],
dist (f,Πm, [a, b]) = minP∈Πm
‖f − P‖∞;[a,b] = ‖f −Bm,a,b‖∞;[a,b]. (11)
The notation [a, b] shall be omitted in case of a = −1, b = 1.
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Proposition 1 (a) Let k ≥ 1 be an integer, m ≥ k be an integer, and f ∈Ck[−1, 1]. Then
dist (f,Πm) ≤ cm−kdist (f (k),Πm−k). (12)
Moreover, for x ∈ [−1, 1],
|f (k)(x)−Bm(k)(f)(x)| ≤ c(
√1− x2 + 1/m)−kdist f (k),Πm−k), (13)
where c is a constant depending on k alone.(b) Let P ∈ Πm. Then for integer k ≥ 1,
maxx∈[−1,1]
|(1− x2)k/2P (k)(x)| ≤ cmk‖P‖∞,[−1,1]. (14)
Proof. In part (a), (12) is proved in [11, Chapter 7, estimate (6.7), p. 220](needs to be iterated k times), the estimate (13) is proved in [11, Chapter 8,Corollary 4.6, p. 249–250]. Part (b) is proved as [11, Chapter 8, Theorem 7.6,p. 265].
Corollary 1 Let r ≥ 1 be an integer, f ∈ Cr[−1, 1], m ≥ r, P ∈ Πm, and
‖f − P‖∞;[−1,1] ≤ ε. (15)
Then for k = 1, · · · , r and x ∈ [−1, 1],∣∣∣(1− x2)k/2(f (k)(x)− P (k)(x)
)∣∣∣ ≤ cdist (f (k),Πm−k) +mkε. (16)
Proof. We use in order, (13), (14) (with Bm(f)−P in place of P ), (11), (12),and (17) to obtain∣∣∣(1− x2)k/2
Let x0 ∈ R, δ > 0, I = [x0 − δ, x0 + δ]. It is easy to verify that if f(y) =f(x0 + yδ), then f (k)(y) = δkf (k)(x0 + yδ), and hence, that
dist (f (k),Πm−k, I) = δkdist (f (k),Πm−k).
Therefore, the above corollary can be reformulated in the form
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Corollary 2 Let x0 ∈ R, δ > 0, I = [x0 − δ, x0 + δ], r ≥ 1 be an integer,f ∈ Cr[I], m ≥ r, P ∈ Πm, and
‖f − P‖∞;I ≤ ε. (17)
Then for k = 1, · · · , r and x = x0 + yδ ∈ I,∣∣∣(1− y2)k/2(f (k)(x)− P (k)(x)
)∣∣∣ ≤ cdist (f (k),Πm−k, I) + δ−kmkε. (18)
In particular,∣∣∣f (k)(x0)− P (k)(x0)∣∣∣ ≤ cdist (f (k),Πm−k, I) + δ−kmkε
. (19)
A multivariate version of Corollary 2, especially the implication that anestimate of the form (17) implies (19) is also immediate, and will be assumednow.
We now revert to the notation in the proof of Theorem 2.2(b), in [1]. A fewtypos are in order. One is that K = [x0 − δ,x0 + δ]q, and subscript K on line 3from below on p. 20 is replaced by [−1, 1]q. Writing
Q(x0 + δy) = Vr(f ,y) = Vr(P,y),
the estimate on line 3 from below on p. 20 (as amended) becomes
‖f −Q‖∞,K ≤ cδs−q/p, ‖P∗n −Q‖∞,K ≤ cδs−q/p.
We now use multivariate version of Corollary 2 with 2r in place of m, Q in placeof P , once with f and once with P∗n to conclude that
|Dk(f −Q)(x0)| ≤ c
dist (Dk,Πq2r−‖k‖1 ,K) + δ−‖k‖1r‖k‖1δs−q/p
, (20)
and
|Dk(P∗n −Q)(x0)| ≤ c
dist (DkP∗n,Πq2r−‖k‖1 ,K) + δ−‖k‖1r‖k‖1δs−q/p
, (21)
Since Dkf and DkP∗n are both in Bc,s−q/p−‖k‖1,∞, we estimate their degrees ofapproximation as in the proof of Theorem 2.2(b), and deduce that
|Dk(f − P∗n)(x0)| ≤ cδs−q/p−‖k‖1 , (22)
where the constant c now depends also on r, which is the integer part of s.
7 Numerical Results
In order to test the above approach, we consider PDEs in various geometries.Starting with a square region, we morph the boundary points using the transfor-mation in (23) to produce a family of boundary curves corresponding to varying
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0.8 0.6 0.4 0.2 0.0 0.2 0.4 0.6 0.80.8
0.6
0.4
0.2
0.0
0.2
0.4
0.6
0.8Geometry corresponding to p=1
Interior PointBoundary Point
0.6 0.4 0.2 0.0 0.2 0.4 0.60.6
0.4
0.2
0.0
0.2
0.4
0.6Geometry corresponding to p=2 Interior Point
Boundary Point
0.6 0.4 0.2 0.0 0.2 0.4 0.60.6
0.4
0.2
0.0
0.2
0.4
0.6Geometry corresponding to p=20
Interior PointBoundary Point
Figure 4: Various geometries generated using a transformation from the Squaregeometry
values of 1 ≤ p ≤ 20. For our purposes we choose p = 1, 2, 20, corresponding tothe geometries shown in Figure 7.
xmorphed = x‖x‖p‖x‖2 ,x 6= 0 (23)
We equi-sample these geometries and interior points closer to the boundarypoints than 0.5h are discarded, where h is the interior sample spacing.
Results below include the maximum relative error in the solution denotedas ‖e‖∞ = ‖u−u0‖∞
‖u0‖∞ . We measure the relative residue as ‖Fu−g‖∞‖g‖∞ where F
denotes the assembled FD matrix, prior to scaling. g denotes the appropriateright hand side. The maximum number of weights used is denoted as L2. Tomeasure the condition number, we use the statistical estimation method of Lauband Kenney [12]. The condition number thus estimated is denoted as κ(F ),where F denotes the FD matrix after scaling. We compare our results with theMatlab FEM Toolbox (PDETool) [13] and Deal II [4].
7.1 Negative Definite Helmoltz
∇2u− u = f
Ω : [−0.5, 0.5]2 ⇒ p
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We assume a complicated solution to the above equation, and set f using u asbelow.
u =1
1 + 1000(x2 + y − 0.3)2+
11 + 1000(x+ y − 0.4)2
+1
1 + 1000(x+ y2 − 0.5)2+
11 + 1000(x2 + y2 − 0.25)2
.
Note that u has singularities very close to the real plane throught the domain.We compare the result of solving this equation with PDETool. We see that ourmethod compares very favourably against this in terms of error achieved withcomparable number of triangles and samples.
Figure 5: Front View of u Figure 6: Top View of u
The following observations are immediate from Figures 7.1 through 9. TheMSNFD solution converges under the arbitraty geometries considered with equi-sampled interior points. The method achieves nearly 4 orders higher accuracycompared with the Matlab FEM method for a comparable number of samples.Also, increasing window size is observed to produce higher order of convergence.The condition number scaling is seem to be similar to that of the FEM approach.There seems to exists a large constant scaling the condition number. The timingchart at the first glance appears to be in favour of the FEM approach. But webelieve that this is misleading. We feel that in order for the comparison to befair, the minimum time taken by each method to get to a given accuracy needsto be compared. The times presented here are both on comparable CPUs. Weextrapolate the time taken by the FEM method to a comparable accuracy. Fig-ure 7 depicts this comparison. This extrapolation was necessary since the FEMmethod could not be run in the computer with the given amount of memory.
Figure 9: Error and Condition number to equation in section 7.1, p = 2
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0 500 1000 150010 8
10 6
10 4
10 2
100
102
1h
!u"
u0! !
!u0! !
Maximum solution error with p = 20
9 ! 9 window11 ! 11 window13 ! 13 windowFEM
0 500 1000 1500102
103
104
105
106
107
108
109
1h
Con
dit
ion
Num
ber
Condition Number with p = 20
9 ! 9 window
11 ! 11 window13 ! 13 window
FEM
Figure 10: Error and Condition number to equation in section 7.1, p = 20
7.2 Negative Definite Helmoltz with Smooth Solution
We now consider the same problem as in the previous solution, but we choosethe right hand side to arise from a rather smooth Runge function as the solution.This example is chosen to depict that the MSNFD method indeed gives backall 12 digits of accuracy with just a 200× 200 grid.
∇2u− u = f
Ω : [−0.5, 0.5]2 ⇒ p
u =1
1 + x2 + y2
7.3 High-Frequency Helmoltz
We first consider a Helmoltz equation with a high wave number. Results ofsolving this problem with the MSNFD method and the Matlab FEM Toolbox(PDETool) are presented in Figures 5 through 8. This being a stiff problem,PDETool is seen to have a lower convergence rate compared to the MSNFD.Considering the minimum time taken to get to a given accuracy, we see thatunless the expected accuracy is less than 2 digits, we see that the MSNFD out-performs the FEM approach. We extrapolate the time take by FEM to get tothe given accuracy. This extrapolation was necessary since the FEM methodcould not be run on our computer with the given amount of memory. For p = 2this minimum sampling rate for the solution to begin to converge is observedto be about 100× 100. While the condition number scales similar to the FEMmethod, there is a large constant scaling it.
∇2u+ 10000u = f
15
0 200 400 600 800 1000 12000
500
1000
1500
2000
2500
1h
Tim
e(s)
Time taken to solve the grand system for Problem 1, with p = 1
Figure 17: Error and Condition number to equation in section 7.3, p = 1
0 500 1000 150010 6
10 5
10 4
10 3
10 2
10 1
100
101
102
1h
!u"
u0! !
!u0! !
Maximum solution error with p = 2
9 ! 9 window11 ! 11 window13 ! 13 windowFEM
0 500 1000 1500102
103
104
105
106
107
108
109
1h
Con
dit
ion
Num
ber
Condition Number , with p = 2
9 ! 9 window
11 ! 11 window13 ! 13 window
15 ! 15 window
FEM
Figure 18: Error and Condition number to equation in section 7.3, p = 2
1 2 3 4 5 610 1
100
101
102
103
104
105
106
107Minimum time taken to get to fixed accuracy with p = 2
log 1!req
t min
MSNFD
FEM
1 2 3 4 5 6102
103
104Minimum grid size to get to fixed accuracy with p = 2
log 1!req
! 1 h
" min
MSNFDFEM
Figure 19: Minimum Time taken to solve the Sparse System from equation insection 7.3 to a given accuracy
19
0 500 1000 150010 8
10 6
10 4
10 2
100
102
1h
!u"
u0! !
!u0! !
Maximum solution error with p = 20
9 ! 9 window11 ! 11 window13 ! 13 windowFEM
0 500 1000 1500102
103
104
105
106
107
108
109
1010
1h
Con
dit
ion
Num
ber
Condition Number , with p = 20
9 ! 9 window11 ! 11 window13 ! 13 windowFEM
Figure 20: Error and Condition number to equation in section 7.3, p = 20
Ω : [−0.5, 0.5]2 ⇒ p
u =1
1 + 1000(x2 + y − 0.3)2+
11 + 1000(x+ y − 0.4)2
+1
1 + 1000(x+ y2 − 0.5)2+
11 + 1000(x2 + y2 − 0.25)2
7.4 Positive Definite Helmoltz - comparison with dealii
In this section, we compare the result of solving an example problem provided bythe Deal II [4] package. It considers a positive definite Helmoltz equation withmixed boundary conditions. f is obtained by using u in the equation below.
Let Γ1 = x = 1 ∩ Ω ∪ y = 1 ∩ Ω and Γ2 be the remaining two sides ofthe square defined by Ω. We use dirichlet boundary conditions on Γ1 given byu, and neumann boundary conditions on Γ2 given by the normal derivatives tou. Figure 12 plots the maximum error in the solution as a function of the griddensity. The graph provides convergence results for varying window sizes forMSNFD and varying element orders for dealii.
7.5 Region with a hole
We consider a region as described below. Table 2 in Figure 11 shows that theMSNFD approach provides very good results for the considered smooth solution.
5.2. Positive Definite Helmoltz - comparison with dealii
In this section, we compare the result of solving an example problem provided by the Deal II [7] package. Itconsiders a negative definite Helmoltz equation with mixed boundary conditions. f is obtained by using u in theequation below.
Let Γ1 = x = 1 ∩ Ω ∪ y = 1 ∩ Ω and Γ2 be the remaining two sides of the square defined by Ω. We use dirichletboundary conditions on Γ1 given by u, and neumann boundary conditions on Γ2 given by the normal derivatives tou. Figure 5.2 below plots the maximum error in the solution as a function of the grid density. The graph providesconvergence results for varying window sizes for MSNFD and varying element orders for dealii.
Table 2: Numerical results for the problemin section5.3, p = 1
5.3. Region with a hole
We consider a region witha hole as shown in Figure 5.3 below. Table ?? below shows that the MSNFD approachprovides very good results for the considered smooth solution. It may also be observed that our simple heuristic ofremoving interior points too close to the boundary points alleviates the geometry induced ill-conditioning.
∇2u − u = f
Ω : [−0.5, 0.5]2\[−0.25, 0.25]2 ⇒ p
u =1
1 + x2 + y2
Figure 21: Numerical results for problems 7.6 and 7.5
Figure 22: Plot of ‖e‖∞ = ‖u−u0‖∞‖u0‖∞ Vs the grid density 1
h
21
It may also be observed that our simple heuristic of removing interior points tooclose to the boundary points alleviates the geometry induced ill-conditioning.
∇2u− u = f
Ω : [−0.5, 0.5]2\[−0.25, 0.25]2 ⇒ p
u =1
1 + x2 + y2
7.6 Multi Scale
We finally consider an example of a PDE whose analytic solution is not known.This is an example of a Multiscale problem as investigated by Shu et al. [14].In their work a multiscale discontinuous Galerkin method was setup to solvethis problem. A spectral method as a reference and convergence results wereobtained. In our case, as in Table 1 in Figure 11, the error measured at agrid density i is the error between the solution at density i and density i − 1when interpolated to the new set of high density grid points, relative to thesolution generated by the grid at density i− 1. These errors were measured byreinterpolating on a 600x600 grid. It may be observed that there is convergenceand the solution is consistent as observed to about 4 digits. Increasing rate ofconvergence is also observed with increasing stencil sizes.
∇.A∇u = x+ y
A =
[1
4+x+sin(x/0.01) 00 1
4+y+sin(y/0.01)
]Ω : [−1, 1]2
u = 0 : ∂Ω
8 Conclusions and Extensions
A Higher Order numerical method to solve elliptic PDEs in two dimensions waspresented. Comparison with Matlab FEM Toolbox (PDETool) as well as Deal IIwere presented. The MSNFD approach performed better and exhibited higherorder of convergence than the FEM methods for problems considered. We be-lieve that a higher order method would be very useful to get to accuracies whichwe believe may be hard and perhaps even impossible with lower order methods.Several additional problems and the corresponding results will be available atour website [5] in the technical report [10]. We believe adaptive gridding to beimportant and this currently work in progress. A comaprable Finite Elementapproach together with the application of MSNFD to the BiHarmonic Typeequations as well as Exterior problems would be presented in our future work[15]. A similar discretization of the integral equation of the PDE and a fastsolution of the resulting dense matrix is possible as well.
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[12] C. S. Kenney, A. J. Laub, M. S. Reese, Statistical condition esti-mation for linear systems, SIAM J. Sci. Comput. 19 (1998) 566–583.doi:10.1137/S1064827595282519.URL http://portal.acm.org/citation.cfm?id=289762.289838
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