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International Journal for Numerical Methods in Engineering, 75, pp. 1607-1630,
2008-02-25
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A new stabilized finite element method for reaction-diffusion problems:
The Source Stabilized Petrov-Galerkin Method
Ilinca, F.; Hétu, J. -F.
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https://nrc-publications.canada.ca/eng/view/object/?id=7bc30fb6-336b-4fee-a5c0-35a2ae0d943d https://publications-cnrc.canada.ca/fra/voir/objet/?id=7bc30fb6-336b-4fee-a5c0-35a2ae0d943d
Published online 25 February 2008 in Wiley InterScience (www.interscience.wiley.com). DOI: 10.1002/nme.2324
A new stabilized finite element method for reaction–diffusion
problems: The source-stabilized Petrov–Galerkin method
F. Ilinca
∗, †, ‡and J.-F. H´etu
‡National Research Council, 75 de Mortagne, Boucherville, Que., Canada J4B 6Y4
SUMMARY
This paper proposes a new stabilized finite element method to solve singular diffusion problems described by the modified Helmholtz operator. The Galerkin method is known to produce spurious oscillations for low diffusion and various alternatives were proposed to improve the accuracy of the solution. The mostly used methods are the well-known Galerkin least squares and Galerkin gradient least squares (GGLS). The GGLS method yields the exact nodal solution in the one-dimensional case and for a uniform mesh. However, the behavior of the method deteriorates slightly in the multi-dimensional case and for non-uniform meshes. In this work we propose a new stabilized finite element method that leads to improved accuracy for multi-dimensional problems. For the one-dimensional case, the new method leads to the same results as the GGLS method and hence provides exact nodal solutions to the problem on uniform meshes. The proposed method is a Galerkin discretization used to solve a modified equation that includes a term depending on the gradient of the original partial differential equation. Copyright q 2008 John Wiley & Sons, Ltd.
Received 15 August 2007; Revised 14 January 2008; Accepted 15 January 2008
KEY WORDS: stabilized finite elements; source stabilization; Taylor series expansion; Petrov–Galerkin; oscillation-free solutions
1. INTRODUCTION
The finite element method is used routinely by scientists to obtain numerical solutions to engi-neering problems. The most common choice of weak form used for finite elements discretization is the Galerkin formulation. The limits of the Galerkin method were first observed for convective-dominated problems for which the finite element solution presents unphysical oscillations. This behavior was the source of an effervescent research on what is now called stabilized finite element methods. Pioneering work was the one by Hughes and Brooks[1] who introduced the concept of upwind discretization for finite elements. The initial approach was further modified to obtain
∗Correspondence to: F. Ilinca, Industrial Materials Institute, National Research Council, 75 de Mortagne, Boucherville,
Que., Canada J4B 6Y4.
†E-mail: florin.ilinca@cnrc-nrc.gc.ca ‡Research Officer.
the streamline upwind Petrov–Galerkin (SUPG) method[2]. This approach deals with convection-dominated problems and contains the key ingredient of a large part of stabilization methods. Similar results can be obtained by the Galerkin least-squares (GLS) method[3], which adds a least-squares term to the Galerkin method.
For instance, when solving transient heat conduction problems with small Fourier number, the Galerkin formulation may fail to produce smooth solutions even if diffusion is the sole transfer mechanism. For example, the one-dimensional unsteady conduction problem in a semi-infinite solid is a well-established case in which such failures can be reproduced[4]. If the thermal boundary layer is thinner than the mesh element size, the Galerkin method produces solutions plagued by spurious oscillations.
To avoid oscillations, Franca and Dutra Do Carmo [5] propose to use the Galerkin gradient least-squares (GGLS) method, which adds stabilization terms to the Galerkin formulation. These terms are obtained by minimizing the square of the residual gradient. In this way, the formulation satisfies the heat balance and at the same time enforces more regularity to the solution gradient.
The GLS formulation was developed as a generalization of the SUPG method, but has recently seen extensive developments in the field of time-harmonic acoustics problems. The singular diffu-sion problem is a particular case corresponding to evanescent waves. Harari and Hughes[6] and Harari et al.[7] applied GLS to the particular case of a constant source term and Dirichlet boundary conditions. For such conditions and assuming very low diffusivity, GLS and GGLS lead to identical solutions and both improve significantly over the Galerkin method. Harari and Hughes [8] and Valentin and Franca [9] combined GLS and GGLS into GLS/gradient least-squares (GLSGLS) in order to provide stability for both convective-dominated problems and problems in which the singular behavior comes from a large undifferentiated term compared with the second-order term. Applications of GLS and GGLS show superiority of one or the other depending on model problem, choice of stability parameters and discretization used. Harari[10] concludes that GLS is superior to GGLS for non-uniform meshes. However, Harari comes to this conclusion solely on a stability analysis for internal nodes and without considering the effect of the boundary conditions. No applications are presented in the paper to support the stability analysis and the ensuing superiority of GLS over GGLS. Harari and Haham[11] compared Galerkin, GLS and GGLS formulations for the solution to elastic wave problems. The paper concludes that the GLS method is not sufficient to correctly handle the directionality of the solution. Alternatively, they mention that the GGLS method yields high accuracy in both magnitude and phase for all directions of propagation. Ilinca and H´etu[4] discussed the issue of global conservation for GLS and GGLS in the context of the singular diffusion problem. They have shown that although GGLS is globally conservative, GLS is not. This causes GLS to produce wrong results when Robin boundary conditions are imposed. In a recent work, Hauke et al.[12] propose to solve advection–diffusion–reaction equations by a subgrid scale/gradient subgrid scale method, which combines two types of stabilization[8, 9]. The resulting formulation provides nodally exact solutions in the one-dimensional case and it recovers the SUPG in the advective–diffusive limit and for linear elements.
In parallel, stabilized methods were proposed by Hughes [13] using a variational multiscale framework and by O˜nate [14] using a finite increment calculus (FIC) approach. The multiscale approach is based on enriching the finite element functions with bubble-like functions describing the unresolvable part of the solution, thus making a link with ‘bubble function’ methods and traditional stabilized methods. Franca et al.[15, 16] developed a multiscale approach for singular perturbed problems by enriching the finite element spaces with local but not bubble-like functions. Two-dimensional benchmark problems show the superiority of their Petrov–Galerkin enriched
method (PGEM) over the residual-free bubbles method [17] and the unusual stabilized finite element method[18].
The FIC techniques are based on a modification of the differential equations to be solved. Although conceptually different, the FIC approach leads to the same finite element equations as the standard SUPG and GLS methods. Ilinca et al.[19, 20] arrive at similar modified differential equations for convective-dominated problems based on a representation of the unresolvable part of the solution. The method was further developed by Ilinca and H´etu[4] to obtain the GLS and GGLS formulations for singular diffusion problems.
In the present work, the solution to reaction–diffusion problems described by the modified Helmholtz operator is discussed and the source-stabilized Petrov–Galerkin (SSPG) method is introduced. This new stabilized finite element formulation improves the accuracy over the Galerkin, GGLS and PGEM methods. The method is shown to be of the Petrov–Galerkin type and consists in modifying the weighting function for the source terms. The stabilized formulation is also shown to be equivalent to a modified equation solved by the Galerkin method. The modified partial differential equation is obtained from the first-order Taylor series expansion around mesh nodes of the terms contained in the original equation. The performance of the SSPG method on one- and two-dimensional problems is illustrated in the numerical results section. The results are compared with those provided by the Galerkin, GGLS and PGEM methods. For the one-dimensional case and a uniform mesh, the new method yields the exact nodal solution as the GGLS formulation. The SSPG is shown to perform better than GGLS and PGEM in the two-dimensional case.
This paper is organized as follows. First, the model problem and the associated Galerkin and GGLS finite element formulations are presented. The SSPG formulation is discussed in Section 3 for the one-dimensional case. Section 4 presents the extension to multi-dimensional problems. Section 5 illustrates the performances of the SSPG method for a selection of one- and two-dimensional test problems. The paper ends with conclusions.
2. THE MODEL PROBLEM
The model problem is described by
L(u)− f = 0 on (1)
u= uD on D (2)
2∇u · ˆn = q on q (3)
where L(u)=2u−2u is the modified Helmholtz operator, is the computational domain, u is the unknown scalar variable, and are physical parameters and f is a given source term. Dirichlet boundary conditions uD are imposed onD and Neumann boundary conditions onq,
withD∩q=∅, and D∪q=* is the boundary of . In Equation (3) q is a given boundary
flux and ˆn is the outward unit vector normal to q.
2.1. The Galerkin formulation
To derive the weak form of this problem, we first introduce the following Hilbert spaces:
V = {u ∈ H1()|u =uD onD} (4) V0= {u ∈ H1()|u =0 on D} (5)
where H1() is the space of functions possessing square-integrable first derivatives in the domain . The variational formulation of this problem is obtained by multiplying Equation (1) by test functions v and integrating by parts the second-order diffusion terms:
Find u∈ V such that 2uvd+ 2∇u ·∇v d− f vd= q qvd ∀v ∈ V0 (6)
Now, let Vhbe a finite dimensional subspace of V . The subspace Vh is generated by a set of C1
piecewise linear basis functions{N1, . . . ,NM} defined on a triangulation T of . We assume that
the triangulation T exactly represents. Hence, any function vh∈ Vhhas the unique representation
vh=MI=1vINI.
The Galerkin finite element formulation can be obtained similarly:
Find uh∈ Vh such that
2uhNId+ 2∇uh·∇ NId− f NId= q q NId ∀NI∈ Vh0 (7) 2.2. The GGLS formulation
When changes in the solution occur at a length scale smaller than the element size, the Galerkin method results in spurious oscillations that are unacceptable for real applications. Such behavior may arise for heat transfer in materials having low conductivity. Very often the solution presents thin thermal boundary layers with a length scale smaller than the element size. The inability of the spatial discretization to correctly represent the solution in the boundary layer generates oscillations[4]. To overcome this problem, Franca and Dutra Do Carmo[5] proposed the GGLS method that adds to the Galerkin equation stabilization terms obtained by minimizing the square of the residual gradient.
The GGLS formulation of the Equation (1) is:
Find uh∈ Vh such that
2uhNId+ 2∇uh·∇ NId− f NId + K K ∇(2uh−2uh− f )·∇∇(2NI)dK= q q NId ∀NI∈ Vh0 (8)
The stabilization terms are integrated only over the elements interior. The definition of the stabi-lization parameter ∇ is (see Reference[5]):
∇= h 2 62 (9) where =cosh( √ 6)+2 cosh(√6)−1− 1 (10) = 2h2 62 (11)
The GGLS formulation has been shown to be nearly optimal for one-dimensional cases. Its extension to multi-dimensional cases also provided significant increase in stability. However, in some cases GGLS still produces detrimental oscillations in the presence of sharp discontinuities in the source term f . In the following sections, we introduce a new stabilized formulation, dubbed SSPG, that produces oscillation-free solutions with improved accuracy compared with the GGLS method.
3. STABILIZATION FOR THE ONE-DIMENSIONAL CASE
3.1. Stability of the standard Galerkin method
One simple way to analyze the stability of the numerical discretization is to consider the one-dimensional problem solved in the domain [0,1]. The source term is taken as zero ( f=0) and the boundary conditions are given by
u(0)= 0 (12)
u(1)= 1 (13)
Following Franca and Dutra Do Carmo[5], this problem has an exact solution given by
u(x )=sinh((/)x)
sinh(/) (14)
The Galerkin finite element discretization using a piecewise linear approximation on a uniform mesh of size h leads to the following equation at node I :
2h
6(uI+1+4uI+uI−1)+
21
h(−uI+1+2uI−uI−1)=0 (15)
or
(uI+1+4uI+uI−1)+(−uI+1+2uI−uI−1)=0 (16)
with given by Equation (11). The contribution of the undifferentiated term can be separated into a pointwise representation and an anti-diffusive term as follows:
(uI+1+4uI+uI−1)=6uI−(−uI+1+2uI−uI−1) (17)
For >1 the anti-diffusive contribution of the undifferentiated term is larger than the natural diffusion of the equation, thus leading to a solution having spurious oscillations. Franca and Dutra Do Carmo reached this observation by expressing the solution to (16) in the form
uI=crI (18)
Substituting (18) into (16) leads to a second-order equation in r :
(1−)r2−2(1+2)r +(1−)=0 (19) having positive roots for <1, but negative roots for >1. In the latter case, the solution has oscillations.
3.2. Limiting the anti-diffusive contribution of the source term
In this work we propose a simple, yet effective method for stabilizing the equation. The prin-ciple is to reduce the anti-diffusive contribution introduced by the Galerkin discretization of the undifferentiated term, which is the source of oscillations in the solution.
Keeping in mind that the basis functions are piecewise linear elements, it is possible to express
uh in a patch of elements surrounding the node I as
uh(x )=uI+
duh(x )
dx (x−xI) (20)
The contribution to the weak form of the undifferentiated term
2uhNId can therefore be decomposed into 2uhNId= 2uINId+ 2duh dx (x−xI)NId (21)
The first term in the right-hand side of Equation (21) is the pointwise representation corresponding to the diagonal term in Equation (17), whereas the second one gives the anti-diffusive term in Equation (17). Stability can therefore be improved by adding to the Galerkin discretization a term that would compensate in part for the contribution of the anti-diffusive term.
We propose to consider a stabilization term of the following form:
−
2duh
dx (x−xI)NId (22)
where is a mesh-dependent stabilization parameter.
In the one-dimensional test case, the solution to the stabilized method is nodally exact when =: =cosh( √ 6)+2 cosh(√6)−1− 1 (23)
In the case of low diffusion (≫1), the stabilization parameter =1 and we recover a pointwise representation of the undifferentiated term also known as ‘mass lumping’.
A similar discretization is used for the diffusion and source terms leading to the following finite element formulation:
Find uh∈ Vh such that
2uhNId+ 2duh dx dNI dx d− f NId − d dx 2uh−2 d2uh dx2 − f (x−xI)NId= q q NId ∀NI∈ Vh0 (24)
4. THE SSPG METHOD FOR THE MULTI-DIMENSIONAL CASE
4.1. Linear shape functions
The extension of the SSPG formulation to multi-dimensional problems is straightforward. For linear interpolation functions, the first-order Taylor series expansion of uh around the node I is
uh=uI+∇uh·(x−xI) (25)
where x=(x, y, z) is the position vector of the point where uhis evaluated, and xI=(xI,yI,zI)is
the position vector of node I . The stabilized finite element formulation in the multi-dimensional case is:
Find uh∈ Vh such that
2uhNId+ 2∇uh·∇ NId− f NId − I∇(2uh−2uh− f )·(x−xI)NId= q q NId ∀NI∈ Vh0 (26) where I=cosh( √ 6I)+2 cosh(√6I)−1− 1 I (27) I= 2h2I 62 (28)
and hI=1/|∇ NI| is of the order of the element size.
Remark 1
The stabilized finite element method is equivalent to solving the modified partial differential equation by a Galerkin method:
[L(u)− f ]−I∇[L(u)− f ]·(x−xI)=0 (29)
As can be seen, stabilization adds to the Galerkin method a term depending on the gradient of the equation residual similar to the method proposed by Ilinca et al.[20] and the FIC method [21]. In this way, the finite element method remains residual in the sense that the exact solution to the partial differential equation is also a solution to the weak formulation. Note, however, that although in the FIC method the gradient of the equation residual is multiplied by the element size, in the present approach it is multiplied by a local position vector originated on the node I to which the equation and the test function NI are associated. Therefore, one element contribution changes
depending on the node to which the test function is associated.
Remark 2
The stabilization term behaves like a diffusion with a diffusion velocity I2(xI−x). Figure 1
shows the apparent diffusion velocity at a point P(x) for the case of one- and two-dimensional elements (in Figure 1 the coefficient I2was neglected and only the vectors xI−x were shown).
(a) (b)
Figure 1. SSPG apparent diffusion velocity for (a) one-dimensional and (b) two-dimensional elements.
Remark 3
Global conservation enforces the variation of the computed quantity (say the energy when solving the energy equation) inside the computational domain to be equal to the sum of fluxes across boundaries and the generation by source terms inside the volume. In the absence of Dirichlet boundary conditions, this identity must be recovered from the finite element formulation when equations for all test functions are added. When Dirichlet conditions are imposed, global conser-vation is reached via auxiliary fluxes (see Hughes et al. [22]) obtained by solving a modified primal–dual formulation. For the Galerkin method, this modified formulation is
2uhNId+ 2∇uh·∇ NId− f NId= * qhNId (30)
where NI denotes test functions in the complete space of the interpolation function Vh, including
points where Dirichlet boundary conditions are imposed [22]. The flux qh is either the flux q
imposed as a boundary condition onq or the auxiliary flux on the boundary containing nodes
with Dirichlet condition. The test functions associated with Dirichlet nodes are used to evaluate the auxiliary flux. This operation can be performed as a post-processing calculation once the solution
uh is obtained by solving Equation (7). Global conservation is reached by adding Equations (30)
for all test functions to obtain 2uhd= f d+ * qhd (31)
The term in the left-hand side measures the global change inside the computational domain, whereas on the right-hand side we have the sum of the production inside the volume and the flux across the boundary.
Similarly, the modified formulation for the stabilized method is 2uhNId+ 2∇uh·∇ NId− f NId − I∇(2uh−2uh− f )·(x−xI)NId= * qhNId (32)
The contribution of the stabilization term to the sum over all test functions is − I∇(2uh−2uh− f )· x M I=1 NI− M I=1 xINI d (33)
but given the fact thatM
I=1NI=1 andMI=1xINI=x, it results in the fact that the contribution
of the stabilization to global conservation is zero. As a result the stabilized formulation is also globally conservative.
Remark 4
The contribution of the diffusion term 2uh to the stabilization is zero for linear elements and
can be neglected.
Remark 5
For the one-dimensional problem solved on piecewise linear elements, the present method gives the same result as the GGLS method. Both solutions are nodally exact on a uniform mesh.
Remark 6
The stabilized equation (26) involves integration of a term containing the differential of the equation residual, which may be discontinuous at the interface between elements if, for example, the source term f is discontinuous. We therefore consider the integral corresponding to the stabilization term as being computed in the sense of distributions
− I∇(2uh−2uh− f )·(x−xI)NId =− K K I∇(2uh−2uh− f )·(x−xI)NId + m= p smp I<2uh−2uh− f >ˆn+·(x−xI)NId (34)
where the first term on the right-hand side of Equation (34) denotes the integral over the element interiors and the second term accounts for the jump between two adjacent elements Km and Kp
that share the common face smp=*Km∩*Kp as shown in Figure 2. The jump operator · has
the property w |smp=w+|smp−w−|smp, where w+=w|Km, w−=w|Kp and ˆn+= ˆn|Km. However,
on all element boundaries, we have either NI=0 or the vector (x−xI)is perpendicular to the
normal vector ˆn and their scalar product is zero (see Figure 2). Therefore, the jump integrals in (34) vanish and the stabilized SSPG equation reduces to a much easier to implement form that contains only integrals over the interior of mesh elements:
2uhNId+ 2∇uh·∇ NId− f NId − K K I∇(2uh−2uh− f )·(x−xI)NId= q q NId (35) Remark 7
The SSPG can be expressed as a Petrov–Galerkin method by performing further manipulations of the stabilization term. As I is constant inside each element, we can introduce the stabilization
Figure 2. Terms on the element boundaries.
parameter inside the differential operator and then integrate by parts to obtain
− K K ∇[I(2uh−2uh− f )]·(x−xI)NId = K K I(2uh−2uh− f )∇ ·(x−xI)NId + K K I(2uh−2uh− f )(x−xI)·∇ NId − K *K I(2uh−2uh− f )(x−xI)· ˆnNId (36)
Here again we use the fact that (x−xI)· ˆnNI=0 on element boundaries *K, and therefore only
the volume integrals over the element interiors remain in Equation (36). These expressions can be further simplified by using
∇ ·(x−xI)=nd (37)
with nd being the dimension of the problem (nd=1 for the one-dimensional case, nd=2 for the
two-dimensional case and nd=3 for the three-dimensional case). We therefore obtain
− I∇(2uh−2uh− f )·(x−xI)NId = K K I(2uh−2uh− f )[ndNI+(x−xI)·∇ NI]d (38)
The resulting stabilized formulation is 2uhNId+ 2∇uh·∇ NId− f NId + K K I(2uh−2uh− f )[ndNI+(x−xI)·∇ NI]d= q q NId (39)
The stabilized method is therefore of the Petrov–Galerkin type: stabilization is equivalent to using the modified test function
NI′= NI+I[ndNI+(x−xI)·∇ NI] (40)
For piecewise linear elements, the Taylor series expansion of NI around xI gives
NI=1+(x−xI)·∇ NI (41)
which means that
(x−xI)·∇ NI= NI−1 (42)
Therefore, the modified test function for source terms takes the form
NI′= NI+I[(1+nd)NI−1] (43)
The shape of the SSPG test function, for I=1, in the one- and two-dimensional cases is illustrated
in Figures 3 and 4, respectively. For the Galerkin method the test function associated with the node I is the same as the shape function. It takes 1 at node I and 0 on the adjacent nodes. The function vanishes on elements not containing node I . The modified test function for SSPG takes a larger value at node I (1+I for one-dimensional problems, 1+2I for two-dimensional problems
and 1+3I for a three-dimensional discretization), and it is equal to −I on nodes adjacent to
node I . This function is also zero on elements not containing node I .
Remark 8
Similar to the PGEM method, the present method is non-symmetric. The PGEM method is based on enriched shape functions that are computed analytically for the case where the
(a) (b)
Figure 3. One-dimensional linear piecewise functions: (a) Galerkin (NI) and (b) SSPG
(a) (b)
Figure 4. Two-dimensional linear piecewise functions: (a) Galerkin (NI) and (b) SSPG
(NI′= NI+[3NI−1]) test functions (=1).
source term f is discretized using the shape functions of the unknown variable. The enriched functions may present very sharp gradients and integrals need to be computed analytically to avoid numerical integration errors. This makes the incorporation into existing finite element solvers as well as the extension to three-dimensional problems more difficult. An alternative to analytical integration was proposed by Oliveira et al. [23] who present numerical inte-gration rules for polynomial times exponential weighting functions. Results are shown for a two-dimensional problem for which the solution is improved by the use of the exponential-adaptive integration formula. One drawback for such a numerical integration is that integration points and integration weights depend on the argument of the exponential function, thus being different from one element to another. The SSPG method involves only piecewise polynomial terms of the same order as those in the Galerkin formulation. The integrals can be evaluated easily using Gaussian quadratures and integration into existing finite element softwares is straightforward.
4.2. Bilinear shape functions
For linear elements, the SSPG (Equation (26)) can be applied directly to two-dimensional triangles and three-dimensional tetrahedra. However, in the case of 4-node quadrilaterals (8-node hexa-hedra for three-dimensional applications) the method has to account for the fact that second-order derivatives of the finite element solution are non-zero. Here we present the case of the bilinear interpolation functions used for the 4-node quadrilateral elements. In this case, the Taylor series expansion of uh around node I is
where∇2uh and (x−xI)2are matrices defined as ∇2uh= ⎡ ⎢ ⎢ ⎢ ⎢ ⎣ *2uh *x2 *2uh *x*y *2uh *x*y *2uh *y2 ⎤ ⎥ ⎥ ⎥ ⎥ ⎦ (45) (x−xI)2= ⎡ ⎣ (x−xI)2 (x−xI)(y− yI) (x−xI)(y− yI) (y− yI)2 ⎤ ⎦ (46)
The stabilized finite element formulation for bilinear elements is therefore 2uhNId+ 2∇uh·∇ NId− f NId − I∇(2uh−2uh− f )·(x−xI)NId − I1 2∇ 2(2u h−2uh− f ):(x−xI)2NId= q q NId (47) 5. NUMERICAL EXAMPLES
The performance of the stabilization method is first verified on the test problems proposed in[5]. Then, a two-dimensional problem having an analytical solution is solved on uniform and non-uniform meshes. Finally, the case of a discontinuous source term resulting in a solution with a sharp gradient inside the computational mesh is shown. The SSPG solution is compared with that provided by the Galerkin, GGLS and PGEM methods.
5.1. One-dimensional test problem
The first problem is a one-dimensional one with the following parameters: 2=1, 2=10−8 and f= x. The problem is solved in a unit domain x ∈[0,1] with boundary conditions u(0)=0 and
u(1)=2. The exact solution would be u = x except for a boundary layer at x =1 where a sharp change is determined by the boundary value. The solutions provided by the Galerkin, GGLS and SSPG methods are compared with the nodally exact solution in Figure 5. As can be seen, the GGLS and SSPG solutions are almost superimposed over the nodally exact solution, whereas the Galerkin solution has oscillations in the boundary layer.
5.2. Two-dimensional test problem with constant source term
The parameters for the two-dimensional problem are 2=1, 2=10−8 and f=1. The equation
Figure 5. One-dimensional problem ( f= x): Galerkin, GGLS and SSPG solutions.
Figure 6. Two-dimensional problem: a uniform mesh.
conditions are
u(0, y)= u(x,0)=0 for 0x, y1 (48)
u(1, y)= u(x,1)=1 for 0<x, y1 (49) The numerical solutions are shown in Figure 7. As observed in Reference[5], the GGLS method eliminates the spurious oscillations that plague the Galerkin solution. However, there is a two-dimensional effect in the boundary layer near (x=0, y =0) that makes the GGLS solution to deviate from the exact solution. The PGEM and SSPG methods result in nodally exact solutions on the entire computational domain.
Figure 7. Two-dimensional problem ( f=1): Galerkin, GGLS, PGEM and SSPG solutions.
The solution is quasi-one-dimensional except near corners. The point where the solution is more sensitive with respect to the finite element discretization is the closest internal node to the corner (x=0, y =0). At this location, the solution changes in both x and y directions. In the present case, the mesh is 20×20; hence, the respective point P has the coordinates x =0.05 and y =0.05 (see Figure 6). The problem was solved using the Galerkin, GGLS, PGEM and SSPG methods and for values of the diffusion coefficient 2ranging from 10−8to 1. The solution was also computed using ‘mass lumping’ for the source terms (MLST), equivalent in finite differences to a pointwise representation of the undifferentiated terms[24]. The results obtained at P(x =0.05, y =0.05) are shown in Figure 8. Note that the solution has to be close to 0 for high values of the diffusion coefficient and then to increase toward 1 as the diffusion coefficient decreases. As can be seen, GGLS, SSPG and MLST provide oscillation-free solutions for all values of the diffusion coefficient (the solution is lower or equal to 1). The Galerkin solution is greater than 1 when 2=10−4 and lower, overestimating the solution by as much as 44%. The PGEM solution is clearly more accurate than the Galerkin one, but it still overestimates the exact solution especially at 2=10−4, whereas the GGLS solution is too diffusive and fails to recover the asymptotic value of 1 as the diffusion coefficient decreases to zero. From all methods, the SSPG and ‘mass lumping’ methods are the most accurate, but the MLST solution seems to be too diffusive for 2=10−4and
Figure 8. Two-dimensional problem ( f=1): Solutions at P(x =0.05, y =0.05).
The problem was also solved with the SSPG method on a mesh formed by 4-node quadrilaterals. The solutions are free of oscillations for all values of the diffusion coefficient. The results shown in Figure 9 for 2=10−8 indicate that the method performs on quadrilateral meshes in the same manner as for triangular meshes.
5.3. Two-dimensional problem with analytical solution
At this point it will be clearly beneficial to solve a problem for which an analytical solution is known; hence, the performance of each method could be compared with respect to the exact solution. For this we solve the same problem but with the following boundary conditions:
u(0, y)=1 2−
sinh((/)(1− y))
2 sinh(/) for 0y1 (50)
u(x ,0)=1 2−
sinh((/)(1−x))
2 sinh(/) for 0x1 (51)
u(1, y)= 1−sinh((/)(1− y))
2 sinh(/) for 0y1 (52)
u(x ,1)= 1−sinh((/)(1−x))
2 sinh(/) for 0x1 (53) having the analytical solution
u(x , y)=1−sinh((/)(1−x)) 2 sinh((/)) −
sinh((/)(1− y))
2 sinh((/)) for 0x, y1 (54) For very low diffusion, the analytical solution is very close to the solution to the previous problem, except on the boundary segment x=0 and on the boundary segment y =0 where the analytical solution is equal to 12 instead of zero (see Figure 10). In the corner (x=0, y =0), both solutions are equal to zero. The results provided by the finite element methods are compared with the exact solution at P(x=0.05, y =0.05) in Figure 11, whereas the errors with respect to the exact
(a) (b)
Figure 9. Two-dimensional problem ( f=1): SSPG solutions for linear and bilinear shape functions: (a) linear shape functions and (b) bilinear shape functions.
Figure 10. Analytical two-dimensional problem: Exact nodal solution for 2=10−8.
solution are plotted in Figure 12. The error of the Galerkin method is much higher than that of the other methods, and hence it was not included in Figure 12. As expected, for very low and very high diffusion coefficients, all stabilized methods perform well. Discrepancies are observed for 2 between 10−6 and 10−2 when the appropriate balance between the natural diffusion and the anti-diffusive contribution of the source term needs to be reached. The results indicate that PGEM overestimates the exact solution (not enough diffusion), whereas the MLST solution is over-diffusive.
A mean nodal error of the solution is estimated by computing the L2 norm of the error with
respect to a nodally exact solution:
en= (uh−uh ex)2d d 1/2 (55)
Figure 11. Analytical two-dimensional problem: Solutions at P(x=0.05, y =0.05).
Figure 12. Analytical two-dimensional problem: Solution errors at P(x=0.05, y =0.05).
where uh=MI=1uINI is the finite element solution and uh ex=MI=1uI exNI is the nodally exact
solution in the space of the finite element shape functions. The mean nodal error is shown in Figure 13 for the various finite element solutions. The SSPG method leads to the most accurate solution for all values of the diffusion coefficient. Note that in the diffusive-dominating limit (large 2) all methods are accurate, but that SSPG and GGLS produce much smaller nodal errors than the Galerkin method. This behavior is caused by the fact that the optimal stabilization parameter as given by Equation (23) does not vanish when →0 and takes the value 12. The stabilization introduced by the GGLS and SSPG methods has the effect of improving the nodal solution even for diffusion-dominated problems.
The problem with analytical solution was also solved on a non-uniform unstructured mesh. In this way, the performance of the finite element methods on non-uniform meshes is verified. The mesh having 213 nodes and 364 triangular elements is shown in Figure 14. The nodal error with respect to the exact solution (see Equation (55)) is shown in Figure 15. Here again SSPG performs very well. The method produces the smallest nodal error for all values of the diffusion coefficient
Figure 13. Analytical two-dimensional problem: Mean nodal error.
Figure 14. Analytical two-dimensional problem: Non-uniform, unstructured mesh.
except for 2=10−1 and 2=1 where PGEM has lower error. The lumped mass technique also provides good results, but its error is constantly higher than that of the SSPG method.
5.4. A two-dimensional problem with discontinuous source term
The SSPG was also tested for the case of a discontinuous source term. This problem verifies the ability of the proposed method to deal with solutions having sharp changes inside the computational domain. The mesh, boundary conditions and parameters are as for the two-dimensional problem treated in Section 5.2. The source term at the mesh nodes is given by
f (xI,yI)= 1 for yI(1+2xI)/4 (56)
Inside a mesh element e, the source term takes the minimum value of the element’s nodal values:
f (x , y)=min
I∈e(f (xI,yI)) for (x, y)∈e (58)
hence, the source term is discontinuous between element meshes as shown in Figure 16. Compu-tations were carried out by using the Galerkin, GGLS, PGEM and SSPG methods and the results are compared in Figure 17 for 2=10−8. As can be seen, the Galerkin solution is plagued by oscillations and is clearly inappropriate. The Galerkin solution varies between−0.375 and 1.375, whereas the exact solution varies between 0 and 1. The GGLS method improves the solution and reduces the oscillations significantly. However, small oscillations are still present and the GGLS solution varies between −0.023 and 1.023. The PGEM and SSPG methods eliminate the oscil-lations and the solution remains between 0 and 1. However, when solving the same problem for 2=10−4, the PGEM is unable to completely avoid oscillations and the solution varies between −0.06 and 1.06 (see Figure 18). The SSPG solution remains oscillation free for all values of
Figure 15. Analytical two-dimensional problem: Mean nodal error on non-uniform, unstructured mesh.
Figure 17. Two-dimensional problem (discontinuous f ): Galerkin, GGLS, PGEM and SSPG solutions for 2=10−8.
the diffusion coefficient. Remark also that the SSPG method does not introduces more artificial diffusion than GGLS. The sharp change in the solution determined by the discontinuity of the source term is actually captured inside a thin layer of two elements between which the source term is discontinuous. This can also be seen in Figure 19 showing the solutions at x=0.45 (also represented by a wider line in Figures 17 and 18). Note that for 2=10−4, the SSPG method is slightly over-diffusive on the elements adjacent to the discontinuity, whereas the PGEM method has oscillations of a comparable magnitude.
6. CONCLUSIONS
A new stabilized finite element method for the singular diffusion problem is presented. The SSPG method is shown to improve the accuracy of the solution over the Galerkin, GGLS, PGEM and lumped mass methods on problems having sharp gradients. It avoids undesirable oscillations resulting in possible unphysical values of the solution.
Numerical experiments on a problem having an analytical solution indicate that the various stabilized methods perform generally well for very high or very low diffusion coefficient. Their
Figure 18. Two-dimensional problem (discontinuous f ): Galerkin, GGLS, PGEM and SSPG solutions for 2=10−4.
(a) (b)
Figure 19. Two-dimensional problem (discontinuous f ): Solutions at
solutions differ, however, inside a wide range of diffusion coefficients corresponding to most practical applications. The PGEM overestimates the exact solution (not enough diffusion), whereas the GGLS and MLST solutions are over-diffusive. The SSPG method leads to the most accurate solution on both uniform and non-uniform meshes, with the exception of the diffusion-dominated case on a non-uniform mesh for which PGEM results in smaller nodal errors. For the case of a discontinuous source term, the SSPG and PGEM methods perform better than the Galerkin and GGLS methods. For moderate values of the diffusion coefficient, SSPG avoids oscillations but it is slightly over-diffusive on the elements adjacent to the discontinuity, whereas the PGEM method has oscillations of a comparable magnitude.
SSPG is equivalent to solving a modified equation by the Galerkin method which includes the first-order derivative of the original partial differential equation. The stabilized method can also be obtained by using a different test function for the source terms. This new stabilized method opens the way for very accurate finite element solutions to the singular diffusion problem.
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![Figure 3. One-dimensional linear piecewise functions: (a) Galerkin (N I ) and (b) SSPG (N I′ = N I +[ 2N I − 1 ] ) test functions ( = 1).](https://thumb-eu.123doks.com/thumbv2/123doknet/14165885.473858/12.850.224.629.716.914/figure-dimensional-linear-piecewise-functions-galerkin-sspg-functions.webp)
![Figure 4. Two-dimensional linear piecewise functions: (a) Galerkin (N I ) and (b) SSPG (N I′ = N I +[ 3N I − 1 ] ) test functions ( = 1).](https://thumb-eu.123doks.com/thumbv2/123doknet/14165885.473858/13.850.225.623.136.430/figure-dimensional-linear-piecewise-functions-galerkin-sspg-functions.webp)
