Local discontinuous Galerkin methods for elliptic problems

Transcription

1 COMMUNICATIONS IN NUMERICAL METHODS IN ENGINEERING Commun. Numer. Meth. Engng 2002; 18:69 75 [Version: 2000/03/22 v1.0] Local discontinuous Galerkin methods for elliptic problems P. Castillo 1 B. Cockburn 1, I. Perugia 2, D. Schötzau 1 1 School of Mathematics, University of Minnesota, 127 Vincent Hall, Minneapolis, MN 55455, USA 2 Dipartimento di Matematica, Università di Pavia, Via Ferrata 1, Pavia, Italy Commun. Numer. Meth. Engng., Vol. 18, pp , 2002 SUMMARY In this paper, we review the development of local discontinuous Galerkin methods for elliptic problems. We explain the derivation of these methods and present the corresponding error estimates; we also mention how to couple them with standard conforming finite element methods. Numerical examples are displayed which confirm the theoretical results and show that the coupling works very well. Copyright c 2002 John Wiley & Sons, Ltd. key words: Finite element methods, discontinuous Galerkin methods, elliptic problems. 1. Introduction Over the last years, discontinuous Galerkin (DG) methods have been successfully applied to a variety of problems where convection phenomena play an important role (see [10] and the references therein for state-of-the-art surveys on DG methods). However, the need to extend these methods to problems in which also diffusion must be taken into account has recently created a renewed interest in DG techniques for elliptic problems; see, e.g., [2, 12, 14]. In this paper, we review the development of one of those methods, namely, the local discontinuous Galerkin (LDG) method. The LDG method was introduced in [11] in the context of convectiondiffusion systems, as generalization of the method proposed in [3] for the compressible Navier- Stokes equations. The LDG method was then further developed and analyzed in [5, 6]; for purely elliptic problems, it has been recently investigated in [4] and [7]. The advantage of applying the LDG method to elliptic problems relies on the ease with which it handles hanging nodes, elements of general shapes, and local spaces of different types; these properties render the LDG method ideally suited for hp-adaptivity and allow it to be easily coupled with other Supported in part by the NSF (DMS ) and by the University of Minnesota Supercomputing Institute. This work was carried out when the author was a Visiting Professor at the University of Minnesota. Copyright c 2002 John Wiley & Sons, Ltd. Received Revised

2 70 LDG METHODS FOR ELLIPTIC PROBLEMS methods. Indeed, the coupling of LDG methods with conforming finite elements (and the performance of the resulting method in the presence of meshes with hanging nodes) is analyzed and numerically tested in [13]; the motivation for this coupling comes from an application in the framework of rotating electrical machines, see [1]. An extension of the LDG method to the Stokes system, where the discretization of the divergence-free condition on the velocity is the main difficulty, was proposed and analyzed in [8]. Extensions to Maxwell s equations are currently under way. 2. The LDG discretization To illustrate the definition of the LDG method, consider the classical model elliptic problem: u = f in Ω, u = g D on Γ D, u n = g N n on Γ N. Here, Ω is a bounded domain in R d, d = 2, 3, and n is the outward unit normal to its boundary Ω = Γ D Γ N. We begin by introducing the auxiliary variable q = u, and by rewriting our model problem as follows: q = u, q = f, in Ω. Let now T be a triangulation of Ω into elements {}; it may have hanging nodes and elements of various shapes. The LDG method determines an approximation (q h, u h ) to (q, u) belonging to the local finite element space Q() U(), typically consisting of polynomials, for each T. This approximate solution is obtained by imposing that, for all (r, v) Q() U(), q h r dx = u h r dx + û h r n ds, q h v dx = fv dx + v q h n ds. Here, n is the outward unit normal to and û h and q h are the so-called numerical fluxes which are discrete approximations to the traces of u and q on the boundary of the elements. The name numerical fluxes is borrowed from the high-resolution schemes for non-linear hyperbolic conservation laws; in that framework, they are also called approximate Riemann solvers. Both numerical fluxes û h and q h are needed in order to achieve a stable and accurate scheme. To define these numerical fluxes, we have to introduce some notation. Let + and be two adjacent elements of T ; let x be an arbitrary point of the set e = + (which is not necessarily an entire edge of an element in T ) and let n + and n be the corresponding outward unit normals at that point. Let (s, w) be a function smooth inside each element ± and denote by (s ±, w ± ) the traces of (s, w) on e from the interior of ±. Then, for a scalar function w and a vector-valued function s, we define the mean values { } and jumps [ ] at x e as {w } := (w + + w )/2, {s } := (s + + s )/2, [w] := w + n + + w n, [s] := s + n + + s n. If e is inside the domain, the numerical fluxes q h and û h are defined by q h := {q h } α[u h ] β[q h ], û h := {u h } + β [u h ],

3 LDG METHODS FOR ELLIPTIC PROBLEMS 71 with a scalar parameter α and a vector-valued parameter β to be properly chosen, whereas on the boundary, we take { { q + h q h := α(u+ h n+ + g D n ) on Γ D, g D on Γ D, and û h := g N on Γ N, u + h on Γ N. This completes the definition of the LDG method. A few important points concerning this method need to be briefly discussed: The aim of the parameters α and β in the definition of the numerical fluxes is to render the formulation stable and to enhance its accuracy. For the method to be well defined, we must have α > 0, whereas β can be arbitrary. The LDG method defines a unique solution under very mild compatibility conditions on the local spaces Q() and U(); in fact, it is enough to have U() Q() (see [4]). The inter-element continuity of the approximate solution as well as the Dirichlet and Neumann boundary conditions are enforced in a weak sense only through the numerical fluxes. The LDG method can be considered to be a mixed finite element method. However, the fact that the numerical flux û h is independent of q h allows us to actually eliminate the variable q h from the equations in an element-by-element fashion. This local solvability gives the name to the LDG method; it is not shared, by classical or stabilized mixed finite element methods. Extensions of the LDG method to more general elliptic problems which include variable diffusion tensors as well as lower order terms can be done in a straightforward way by applying the techniques developed in [6]. 3. Error estimates A complete error analysis of the LDG method has been carried out in the case where the local spaces are taken as standard polynomial spaces with possibly different approximation degrees, Q() = P l () d, U() = P k (), with k 1 and l = k or l = k 1. We discuss the cases l = k and l = k 1 separately, giving rise to equal-order elements and mixed-order elements, respectively. Our theoretical results for the local spaces above, are summarized in Table I below; h denotes the biggest diameter of the elements of the triangulation T. Table I. Orders of convergence of the L 2 -errors in q and u for smooth solutions and k 1. α Q() U() L 2 -error in q L 2 -error in u O(1) P k () d P k () k k + 1/2 O(1/h) P k () d P k () k k + 1 O(1) P k 1 () d P k () k 1/2 k O(1/h) P k 1 () d P k () k k + 1 Equal-order elements. In this case, it has been proved in [4] that the orders of convergence of the L 2 -norms of the errors in q and u are k and k + 1, respectively, provided that the parameters α and β are taken to be of order O(h 1 ) and O(1), respectively. These orders

4 72 LDG METHODS FOR ELLIPTIC PROBLEMS were actually observed numerically in the experiments reported in [4]. For α of order O(1), the theoretical orders of convergence k and k + 1/2 were obtained in [4], resulting in a loss of h in the approximation of u. However, no degradation in the order of convergence for u was actually observed on unstructured triangular meshes in the numerical results reported in [4] and [13]. Furthermore, it has been shown in [7] that on Cartesian grids, with a special choice of the numerical fluxes (for which α is of order O(1) and β is such that β n = 1 2 ) and for equal-order elements with Q k -polynomials, the LDG method super-converges and the orders of convergence k in q and k + 1 in u are obtained. A similar phenomenon has not been observed on unstructured grids. Mixed-order elements. LDG methods with lower approximation degree for q h (i.e., with l = k 1) have been first analyzed in [8] in the context of the Stokes problem. These results immediately carry over to the model problem considered here: with α of order O(1/h) and β of order O(1), the same orders of convergence k in q and k+1 in u as for equal-order elements are achieved. In this case, the error estimates are optimal from both an approximation point of view and in terms of the smoothness requirements on the exact solution. However, numerical tests in [8] and [13] indicate that the use of equal-order elements is not less efficient. Coupling with conforming finite elements. When the LDG methods described above are coupled with the standard conforming finite element method that uses polynomials of degree k, the orders of convergence are identical to those reported in the above paragraphs; see [13]. 4. Numerical experiments Results on an L-shaped domain. First, we report the results of two numerical experiments from [4]. We solve the model problem on an L-shaped domain, using a sequence of unstructured triangular meshes created from consecutive global refinement. The initial coarse mesh consists of 22 elements. The parameter α is chosen as 1/h, and β such that β n = 1 2. Table II. H 5 -solution on L-shaped domain: Orders of convergence of the L 2 -errors in q and u. k L 2 -error in q L 2 -error in u h h/2 h/4 h/8 h h/2 h/4 h/ In Table II, we show the orders of convergence in the L 2 -errors of q and u for the LDG method using equal-order elements of degree k = 1,..., 5, for a solution belonging to H 5 (Ω). The numerical orders are obtained from the errors of two consecutive meshes. The expected orders of min(4, k) and min(5, k + 1), respectively, are clearly visible. In Table III, we report results for the exact solution expressed in polar coordinates by u(r, θ) = r γ sin (γθ), with γ = 2/3, that exhibits a singularity at the reentrant corner of the L-shaped domain. The results show that we get orders close to 2 3 and 4 3. These are the

5 LDG METHODS FOR ELLIPTIC PROBLEMS 73 same orders that are obtained with a standard conforming method, for which the orders of convergence in the H 1 (Ω)- and L 2 (Ω)-norm are 2 3 ε and 4 3 ε, respectively, for all ε > 0. Table III. Singular solution on L-shaped domain: Orders of convergence of the L 2 -errors in q and u. k L 2 -error in q L 2 -error in u h h/2 h/4 h/8 h h/2 h/4 h/ Coupling with conforming finite elements. In the following numerical example, taken from [13], we consider the situation where the LDG method is applied only in a sub-domain Ω LDG Ω and a conforming method is used in the remaining part Ω conf = Ω \ Ω LDG. The corresponding coupled method, introduced in [1] and analyzed in [13], combines the ease with which the LDG method handles hanging nodes with the lower computational cost of standard conforming finite elements. This approach is also very promising in the context of multi-physics or multi-material problems, where the practitioner might want to use a DG method only in certain parts of the computational domain. Figure 1. Meshes used in the numerical experiments: Non-nested grids with 256 and 1024 elements and hanging nodes on the line y = 0. The LDG region Ω LDG is shadowed. The coupling across the common interior interface Γ of Ω LDG and Ω conf is achieved as follows. On the LDG side, the interface Γ is considered as a Dirichlet boundary with datum given by the trace on Γ of the approximation from the conforming side, whereas on the conforming side Γ is considered as a Neumann boundary, with datum given by the corresponding flux from the LDG side; see [13] for details. We solve the model problem on Ω = ( 1, 1) ( 1, 1), using a sequence of non-nested grids with decreasing mesh-sizes with a reduction factor 2. The grids are obtained by meshing independently the two subregions ( 1, 1) ( 1, 0) and ( 1, 1) (0, 1), giving rise to possibly non-matching grids along the line y = 0. We define the LDG region Ω LDG on each mesh as the union of all the elements having at least one vertex on the line y = 0. The domain Ω LDG

6 74 LDG METHODS FOR ELLIPTIC PROBLEMS contains all the hanging nodes and shrinks towards the line y = 0 as h goes to zero; the second and third meshes are depicted in Figure 1. In Table IV, we display the L 2 -errors in q and u obtained when the LDG method with equal-order P 1 -elements is applied in Ω LDG, and standard conforming P 1 -elements are used in Ω conf. The results show that the errors converge with the optimal orders, as expected, and that the coupling of LDG and conforming methods with a shrinking LDG region can be successfully carried out. Table IV. Convergence for the coupled method and P 1 -elements; α = 1, β = O(1). reduction L 2 -norm of q L 2 -norm of u in mesh-size h error reduction error reduction e e e e e e e e Conclusions The results of an error analysis of the LDG method for elliptic problems have been reviewed and numerical experiments have been presented, showing the sharpness of the theoretical estimates. It has also been indicated how to couple LDG and conforming finite element methods, in order to exploit the advantages of the LDG method with a reduced computational cost. REFERENCES 1. P. Alotto, A. Bertoni, I. Perugia, and D. Schötzau, Discontinuous finite element methods for the simulation of rotating electrical machines, COMPEL, 20 (2001), D. Arnold, F. Brezzi, B. Cockburn, and D. Marini, Unified analysis of discontinuous Galerkin methods for elliptic problems, SIAM J. Num. Anal., to appear. 3. F. Bassi and S. Rebay, A high-order accurate discontinuous finite element method for the numerical solution of the compressible Navier-Stokes equations, J. Comput. Phys., 131 (1997), pp P. Castillo, B. Cockburn, I. Perugia, and D. Schötzau, An a priori error analysis of the local discontinuous Galerkin method for elliptic problems, SIAM J. Num. Anal., 38 (2000), P. Castillo, B. Cockburn, D. Schötzau, and C. Schwab, An optimal a priori error estimate for the hp-version of the local discontinuous Galerkin method for convection diffusion problems, Math. Comp., to appear. 6. B. Cockburn and C. Dawson, Some extensions of the local discontinuous Galerkin method for convection-diffusion equations in multidimensions, Proceedings of the X Conference on the Mathematics of Finite Elements and Applications: MAFELAP X, J. Whitemann, ed., Elsevier, 2000, pp B. Cockburn, G. anschat, I. Perugia, and D. Schötzau, Superconvergence of the local discontinuous Galerkin method for elliptic problems on Cartesian grids, SIAM J. Num. Anal., 39 (2001), B. Cockburn, G. anschat, D. Schötzau, and C. Schwab, Local discontinuous Galerkin methods for the Stokes system, SIAM J. Num. Anal., to appear. 9. B. Cockburn, G. arniadakis, and C.-W. Shu, The development of discontinuous Galerkin methods, in Discontinuous Galerkin Methods: Theory, Computation and Applications, B. Cockburn, G. arniadakis, and C.-W. Shu, eds., vol. 11 of Lect. Notes Comput. Sci. Eng., Springer Verlag, 2000, pp B. Cockburn, G. arniadakis, and C.-W. Shu, eds., Discontinuous Galerkin Methods. Theory, Computation and Applications, vol. 11 of Lect. Notes Comput. Sci. Eng., Springer Verlag, 2000.

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