Singularly Perturbed Partial Differential Equations

Transcription

1 WDS'9 Proceedings of Contributed Papers, Part I, 54 59, 29. ISN MTFYZPRESS Singularly Perturbed Partial Differential Equations J. Lamač Charles University, Faculty of Mathematics and Physics, Prague, Czech Republic. bstract. We present a summary of basic numerical methods for solving singularly perturbed partial differential equations. First the effect of the presence of a singular perturbation is introduced, then two different approaches to the elimination of this effect are discussed. Stabilization techniques are demonstrated on several test examples. daptive mesh-refinement methods are taken into account afterwards. Introduction We call a partial differential equation singularly perturbed when its behaviour and properties signifficantly change while one (or more) of its coefficients tends to some critical value. In this paper the singular perturbation will be always caused by the coefficient ε +. The model equation for our purposes will be the scalar convection-diffusion equation: ε u + b u = f in Ω = (,) d, u = u b on Ω. () Here d = or 2, ε > is a constant coefficient of diffusion, b W, (Ω) d is a convective field satisfying the incompressibility condition divb = and f L 2 (Ω) is a given outer force. We also assume that the Dirichlet boundary condition satisfies u b H /2 ( Ω). Since it is usually impossible to find an analytic solution of the problem () for general data, we have to solve it numerically. It is especially difficult when the convection dominates diffusion, i.e. ε b. Then we have a convection dominated problem and the equation becomes singularly perturbed. In this case the solution of () usually possesses interior and boundary layers. These are narrow regions where the solution changes rapidly. If the mesh size is much larger then the width of these regions, the layers cannot be resolved properly and thus spurious (nonphysical) oscillations occur in the numerical solution. To diminish these oscillations one usually uses upwinding or its multi-dimensional analogy SUPG (Streamline upwind Petrov/Galerkin) method. Despite the fact that the SUPG method significantly improves the quality of the discrete solution, it does not remove all spurious oscillations at layers. For this reason another stabilization terms are added to the discrete problem. These are the so-called SOLD (Spurious oscillation at layers diminishing) methods. One can also use some adaptive mesh-refinement algorithm and refine the mesh along layers. However, it does not always bring the desired effect since the mesh width in layer regions should be extremly small. Weak formulation and finite element discretization To employ the finite element method one has to start with a weak formulation of the problem (). Let ũ b H (Ω) be an extension of u b to the whole domain Ω, then the weak formulation of the problem () reads: Find u H (Ω) such that u ũ b H (Ω) and where a(u,v) = (f,v) v H (Ω), (2) a(u,v) = ε( u, v) + (b u,v). (3) Here and throughout this paper (, ) denotes an inner product in L 2 (Ω). Under the assumptions mentioned above the Lax-Milgram theorem gives the existence and uniqueness of the weak solution u H (Ω) [Lamač, 28]. 54

2 LMČ: SINGULRLY PERTURED PRTIL DIFFERENTIL EQUTIONS The starting point of the finite element discretization is the definition of a triangulation T h of the domain Ω. It consists of a finite number of open polygonal elements K (intervals for d = ) and satisfies Ω = K T h K with diam K h, K T h. The intersection of closures of any two different elements K is either empty or a common vertex or a common edge (d = 2). When d = 2 we confine ourselves just to triangular elements with some additional shape-regularity assumption (the minimum angle assumption or bounded ratios of triangle s diameter to its inscribed circle). Now we obtain a finite element discretization of equation (2) replacing the space H (Ω) by a finite-dimensional finite element subspace V h. We say that u h H (Ω) is a discrete solution obtained by the finite element method if u h ũ bh V h and where ũ bh is a finite element interpolate of ũ b. Stabilization methods a(u h,v h ) = (f,v h ) v h V h, (4) Let us start our observations with a D example. Example. Consider equation () with d =, b const. >, f and u b, i.e. εu + bu = in (,), u() = u() =. (5) The analytical solution of the problem (5) is ( u(x) = b x exp(b ε x) exp( b ε ) ). (6) Solving the problem (5) by continuous piecewise linear finite elements leads to the central finite difference scheme and consequently to unwanted spurious oscillations. The usual remedy is using an upwind difference discretization of the convective term. Using the finite element method one can reach the upwind effect by adding an artificial diffusion ε u = 2bh to the diffusion constant ε: ε u i+ 2u i + u i h 2 + b u i u i h ( = ε + bh ) ui+ 2u i + u i 2 h 2 + b u i+ u i. (7) 2h Here and in what follows x i = ih and u i u(x i ), for i =,,...,N and h = /N, N N. Denote α = bh/2ε the local Péclet number and ξ(α) = coth α /α. Then we can obtain a nodally exact solution [Lamač, 28] of the equation (5) by adding the artifical diffusion ε opt = ε u ξ(α) < ε u to the diffusion constant ε in the central difference scheme (see Figure ) Figure. Details of the discrete () and the exact () solution of Example (with b ) at a layer near x = obtained using the finite element method with added artifical diffusion ε u (left) and ε opt (right). nother way how to acquire the stabilization effect is a change of the weighting (test) functions in the discrete problem (4). The standard piecewise linear weighting functions are v i = x x i χ h [xi,x i ] + x i+ x χ h [xi,x i+ ], i =,2,...,N, (8) where χ I denotes a characteristic function of the interval I. 55

3 LMČ: SINGULRLY PERTURED PRTIL DIFFERENTIL EQUTIONS We change these functions by adding some constants on both intervals [x i,x i ] and [x i,x i+ ]. Thus the modified weighting functions have a form (see Figure 2) v i [c L,c R ] = (v i + c L )χ [xi,x i ] + (v i + c R )χ [xi,x i+ ]. (9) If we choose c L = /2 and c R = /2, then the solution obtained using the finite element method is the same as that obtained using the finite difference method with the full upwind discretization of the convective term. If we choose c L = ξ(α)/2 and c R = ξ(α)/2 then we obtain again the nodally exact solution of the equation (5). v i v U i x i+ x i x i x i+ x i x i Figure 2. The standard weighting function v i (left) and the weighting function v Ui = v i [/2, /2] (right) which causes the upwind effect. If we then apply a similar approach of changed weighting functions in higher dimensions (d > ), we obtain the SUPG method [rooks, 982]. Denoting the residual R(u) = ε u + b u f, the SUPG method reads: Find u h H (Ω) such that u h ũ bh V h and a(u h,v h ) + (R(u h ),τb v h ) = (f,v h ) v h V h. () The function τb v h is (in our case) discontinuous and corresponds to the choice of the constants c L,c R in one dimension. The parameter τ is the so-called stabilization parameter and, in D, we obtain the nodally exact solution for τ = h 2bξ(α) [Lamač, 28]. The choice of the stabilization parameter τ L (Ω) is in higher dimensions [John, 27] recommended as: τ K = τ K = µ K h K 2 b ξ(α K) with α K = ν K b h K 2ε. () Here h K is a diameter of the element K in the direction of b and µ K = ν K = /k, where k is the order of approximation of V h K with respect to the norm,k. The SUPG method does not diminish all the spurious oscillations in the discrete solution, particularly at layers. That is why we add another stabilization term to the left hand side of the equation (). These techniques belong to the so-called SOLD (Spurious oscillations at layers diminishing) methods. The most efficient [John, 27] SOLD methods are the do Carmo-Galeão method [Carmo, 99], the urman-ern method [urman, 22] and mainly the Mizukami- Hughes method [Mizukami, 985]. When u h, the do Carmo-Galeão method adds to the left hand side of () the term ( ε u h, v h ), where ε = σ R(u h) 2 and σ = τ max{, b u h u h 2 R(u h ) }. nother SOLD method term was derived by urman and Ern. When b it adds the term ( ε D u h, v h ) to the left hand side of (). Here D = I (b b)/ b 2 is a projection into the plane perpendicular to the vector b and ε = τ b 2 R(u h ) /( b u h + R(u h ) ). While the last two methods add another stabilization term to the equation (), the Mizukami-Hughes method completely changes the weighting functions in (4). It changes the standard piecewise linear weighting functions v i by adding constants Ci K on elements K suppv i : ϕ i = v i + Ci K χ K, v i V h. (2) K supp v i Thus, for each K, we have three constants Ci K corresponding to the vertices of K and it is assumed that their sum vanishes (see Figure 3). The functions ϕ i are then applied to the discrete problem (4). 56

4 LMČ: SINGULRLY PERTURED PRTIL DIFFERENTIL EQUTIONS a) b ) c ) d ) 2/3. v i P i supp v i b.23 2/3 /6 /6 2/3 ϕ i Figure 3. The construction of the weighting function in the Mizukami-Hughes method: for each element K from the support (b) of the standard piecewise linear weighting function v i (a) one computes three constants Cj K coresponding to the vertices of K (c). The constants Ci K corresponding to the vertex P i are then on each element K added to the function v i (d). proper choice of constants Ci K depends not only on the direction of the vector b, but also on the solution u h itself. Thus, as well as the two SOLD methods mentioned above, the Mizukami-Hughes method is nonlinear. etter properties, in particular at layers, has the improved Mizukami-Hughes method (IMH) introduced by Knobloch [Knobloch, 26]. The following example shows a result of an application of both the SUPG and the improved Mizukami- Hughes method in 2D. Example 2. Let us consider the problem () with d = 2, ε = 3, b = (/2, 3/2) T, f and a boundary condition u b = for x = or y.7 and u b = otherwise. If we apply the SUPG method on an isotropic mesh, we obtain a solution with spurious oscillations at layers. s we can see in Figure 4 the improved Mizukami-Hughes method provides a remedy Figure 4. Solutions of Example 2 solved on an isotropic mesh (left, 8 triangles) using the SUPG (center) and the improved Mizukami-Hughes (right) method. different approach that leads to the stabilizing effect is a stabilization via local projection [Roos, 28]. It introduces a projection π h : L 2 (Ω) D h into a second finite element space D h and then defines a stabilization term based on the fluctuation κ h (b u h ) of b u h, where κ h = id L 2 (Ω) π h. The space D h is generally defined on the different triangulation M h in such a way that several assumptions on its dimension and order of approximation are satisfied. In our case of continuous piecewise linear finite elements we define the space D h on the same triangulation as the space V h, but it consists only of piecewise constant functions. The space V h is also changed. It is enriched element by element with bubble functions β K defined as the product of the barycentric coordinates. Then we say that u h H (Ω) is a solution obtained by the local projection stabilization method, if u h ũ bh V h and where the stabilizing term is given by S h (u h,v h ) = a(u h,v h ) + S h (u h,v h ) = (f,v h ) v h V h, (3) K T h τ K (κ h (b u h ),κ h (b v h )) K. (4) 57

5 LMČ: SINGULRLY PERTURED PRTIL DIFFERENTIL EQUTIONS If we choose τ K = τ opt = h 6 (/ξ(α) 3/α) < h 6 on each element K T h, then the obtained solution is in D for constant data (and an equidistant mesh with diam K = h for all K T h ) nodally exact. The result of an application of this metod is depicted in Figure C C C Figure 5. n application of the local projection method to Example. The coefficient τ K is chosen as, τ opt and h/6, respectively. The solution obtained by the local projection method () is compared with the exact solution (C). In order to remove the oscillations bubble functions are omitted afterwards (). daptively refined meshes The strategy of using adaptively refined meshes consists in a localization of those parts of the computational domain Ω where the discrete solution differs the most from the exact solution. In these regions the mesh is refined in order to allow better approximation of the solution. To localize the problematic regions one uses a posteriori error estimators. They usually depend not only on the coefficients of the equation but also on the applied triangulation or computed solution. While local error estimators (indicators) determine regions where the highest error appears, a global error estimator determines the error in the whole computational domain Ω. It can be derived as a sum of local error estimators. Let us assume that we can assign a local error estimator η K to each element K T h and let the global error estimator η be a sum of local error estimators. Then a general mesh-refining algorithm has following steps:. Construction of an initial triangulation T h. 2. Solution of the discrete problem on T hk. 3. Computation of local error estimators η K. 4. Evaluation of a global error estimate (if sufficient then stop). 5. Selection of elements to refine taking η K into account. 6. Construction of a new triangulation, go to 2. very convenient algorithm for the construction of new triangulations (in step 6) is the redgreen algorithm. It divides selected elements in such a way that the shape-regularity (minimum angle) condition is always satisfied, i.e. the angels in each triangle of each triangulation do not degenerate. nother possibility how to costruct new triangulation is moving nodes in the direction of the increasing error. Combining both approaches one can reach satisfactory results Figure 6. Meshes obtained by the red-green algorithm (left) and NGENER (right). They consist of 7833 and 9859 elements, respectively. 58

6 LMČ: SINGULRLY PERTURED PRTIL DIFFERENTIL EQUTIONS The previous pictures show meshes obtained by the red-green algorithm and a program NGENER (see references) applied to Example 2. During the red-green algorithm the a posteriori error estimator introduced by Kunert [Kunert, 22] was used. The program NGENER not only refines the mesh but also moves nodes. Nevertheless the program uses an error estimator which works just with the computed discrete solution and does not depend directly on data of the equation (ε,b,f). Finally we can evaluate the global error estimate (by Kunert) for all types of methods we used while solving the Example 2. The following table confirms our expectations that the global error estimate (computed using the error estimator introduced by Kunert) decreases much more rapidly when mesh-refinement algorithms are applied. Meshes from Figures 4 and 6 were used. Method SUPG IMH RED-GREEN NGENER NGENER+IMH Error estimate Discussion We have presented several efficient stabilization techniques for solving singularly perturbed partial differential equations and compared them on test examples. If we use only stabilization methods we cannot achieve a sufficiently small global error (unless we use large amount of nodes). On the other hand in a strongly convection dominated case we cannot diminish all spurious oscillations at layers using only adaptive mesh-refinement since the layers are too narrow. In higher dimensions it is probably important to construct such triangulations whose elements are thinner in the direction where the oscillations are observed. Due to an added artificial diffusion, stabilization methods can also cause smearing of the solution near layers. Conclusion ll experiments confirmed the fact that singularly perturbed partial differential equations are numerically very difficult to solve. If we want to solve them successfully, we have to appropriately combine stabilization and adaptive methods in order not only to diminish all spurious oscillations at layers but also acquire sufficiently accurate solution. References NGENER 3., mesh generator and mesh adaptor, version 3., rooks,. N., T. J. R. Hughes, Streamline upwind/petrov-galerkin formulations for convection dominated flows with particular emphasis on the incompressible Navier-Stokes equations, Computer Methods In pplied Mechanics nd Engineering, 32, , 982. urman, E., E. Ern, Nonlinear diffusion and discrete maximum principle for stabilized Galerkin approxiations of the convection-diffusion-reaction equation, Computer Methods In pplied Mechanics nd Engineering, 9, , 22. Carmo, E. G. D. do,. C. Galeão, Feedback Petrov-Galerkin methods for convection-dominated problems, Computer Methods In pplied Mechanics nd Engineering, 88, -6, 99. John, V., P. Knobloch, comparison of spurious oscillations at layers diminishing (SOLD) methods for convection-diffusion equation: Part I - review, Computer Methods In pplied Mechanics nd Engineering, 96, , 27. Knobloch, P., Improvements of the Mizukami-Hughes method for convection-diffusion equations, Computer Methods In pplied Mechanics nd Engineering, 96, , 26. Kunert, G., posteriori error estimation for convection-dominated problems on anisotropic meshes, Preprint- Reihe des Chemnitzer Sonderforschungbereich 393/2-4, Technische Universität Chemnitz, March, 22. Lamač, J., Numerical solution of convection-diffusion equations using stabilization and adaptive methods, Master s Thesis, Charles University in Prague, 28. Mizukami,., T. J. R. Hughes, Petrov-Galerkin finite element method for convection-dominated flows: n accurate upwinding technique for satisfying the maximum principle, Computer Methods In pplied Mechanics nd Engineering, 5, 8-93, 985. Roos, H.-G., M. Stynes, L. Tobiska, Robust Numerical Methods for Singularly Perturbed Differential Equations, Springer-Verlag erlin Heidelberg,