A Balancing Algorithm for Mortar Methods Dan Stefanica Baruch College, City University of New York, NY, USA. Dan_Stefanica@baruch.cuny.edu Summary. The balancing methods are hybrid nonoverlapping Schwarz domain decomposition methods from the Neumann-Neumann family. They are efficient and easy to implement. We present a new balancing algorithm for mortar finite element methods. We prove a condition number estimate which depends polylogarithmically on the number of nodes on each subregion edge and does not depend on the number of subregions of the partition of the computational domain, just as in the conforming finite element case. Introduction The balancing method of Mandel [7] is a hybrid nonoverlapping Schwarz domain decomposition method from the Neumann-Neumann family. It is easy to implement and uses a natural coarse space of minimal dimension which allows for an unstructured partition of the computational domain. The condition numbers of the resulting algorithms depend polylogarithmically on the number of degrees of freedom in each subregion. There is a close connection between the balancing method and the FETI [5] and FETI DP [4] methods; cf. [6]. A new version of the balancing method, also related to FETI type algorithms, was recently proposed by Dohrmann et al. [9, ]. Mortar finite elements were first introduced by Bernardi et al. [] and are actively used in practice because of their advantages over the conforming finite elements, e.g., flexible mesh generation and straightforward local refinement. In this paper, we propose an extension of the balancing method to mortar finite elements. As in conforming cases, every local space is associated with a subregion from the partition of the computational domain. The values of the mortar function on a nonmortar side depend on, but are not equal to, its values on the mortar sides opposite the nonmortar. To account for this dependence, the local spaces are defined on extended subregions, instead of using local spaces and local solvers defined on each subregion. In this regard, our algorithm is different from classical Neumann-Neumann methods.
748 D. Stefanica We establish a polylogarithmic upper bound for the condition number of our algorithm. The same bound has been obtained for the balancing algorithm of Dryja [], as well as for other mortar algorithms, e.g., the iterative substructuring method of Achdou et al. [], in the geometrically nonconforming case. While the algorithm proposed here is based on a similar philosophy as the method suggested in [], since the Schwarz framework is used to study the convergence properties of both algorithms, major differences exist between the two algorithms. For example, in the algorithm of [], the local spaces are associated with pairs of opposite nonmortar and mortar sides. Abstract Schwarz Theory We use this elegant framework of the abstract Schwarz theory [] to study the convergence properties of the balancing algorithm proposed in this paper. Let V be a finite dimensional space, with a coercive inner product a : V V R,andlet f : V R be a continuous operator. We want to find the unique solution u V of a(u, v) = f(v), v V. () Assume that V can be written as V = V +V +...+V N, where the sum is not necessarily direct nor do we necessarily have V i V, i =:N.Let I i : V i V be embedding operators and let ã i : V i V i R be bilinear forms which are symmetric, continuous, and coercive. The corresponding projection-like operators T e i : V V i are defined by ã i( T e iv, v i) = a(v, I iv i), v i V i, v V. () Using the operators T i : V V, T i = I i et i, the additive and multiplicative Schwarz methods for solving () can be introduced. The balancing method is a hybrid method, combining the potential for parallelization of the additive methods and the fast convergence of the multiplicative methods. Choose the bilinear form ã to be exact, i.e., ã (, ) =a(, ). The coarse space solver T is therefore a projection, subsequently denoted by P. The balancing method consists of solving T bal u = g bal,where T bal = P +(I P )(T + + T N)(I P ). () Here, g bal is obtained by solving N local problems of the same form as () that do not require any knowledge of u. The equation T bal u = g bal is a preconditioned version of () and can be solved without further preconditioning using CG or GMRES algorithms. A Mortar Discretization of an Elliptic Problem As a model problem for two dimensional self adjoint elliptic PDEs with homogeneous coefficients, we choose the Poisson problem with mixed boundary conditions on Ω :given f L (Ω), find u H (Ω) such that u = f on Ω, with u =on Ω D and u/ n =on Ω N, (4)
A Balancing Algorithm for Mortar Methods 749 where Ω N and Ω D are the parts of Ω = Ω N Ω D where Neumann and Dirichlet boundary conditions are imposed, respectively, and Ω D has positive Lebesgue measure. To keep the presentation concise, we only discuss geometrically conforming mortar elements. Let {Ω i} i=:n be a geometrically conforming mortar decomposition of a polygonal domain Ω of diameter into rectangles of diameter of order H. (This notation is not coincidental: for the balancing method proposed here, each local space V i will correspond to one subregion Ω i.) The restriction of the mortar finite element space V h to any rectangle Ω i is a Q finite element function on a mesh of diameter h. Weakcontinuityisrequiredacross Γ, the interface between the subregions {Ω i} i=:n. We choose a set of edges of {Ω i} i=:n, called nonmortars, which form a disjoint partition of Γ. For each nonmortar side γ there exists exactly one side opposite to it, which is called a mortar side. The jump [w] of a mortar function w V across any nonmortar γ must be orthogonal to a space of test functions Ψ(γ), i.e., [w] ψds =, ψ Ψ(γ). (5) γ In [], Ψ(γ) consists of continuous, piecewise linear functions on γ that are constant in the first and last mesh intervals of γ. Note that the end points of the nonmortar sides are associated with genuine degrees of freedom. We discretize the Poisson problem (4) by using the mortar finite element space and obtain the discrete problem: V h find u h V h such that a Γ (u h,v h )=f(v h ), v h V h, (6) where the bilinear form a Γ (, ) is defined as the sum of contributions from the individual subregions, and f( ) is the L -inner product by the function f : a Γ (v h,w h ) = v h w h dx and f(v) = Ω i i= Ω fv dx. Let V h (Γ ) be the restriction of V h to the interface Γ,andlet V be the space of discrete piecewise harmonic functions defined as follows: if v Γ V h (Γ ), then its harmonic extension H(v Γ ) V is the only function in V h which, on every subregion Ω i, is equal to the harmonic extension of v Γ Ωi with respect to the H -seminorm. As in other substructuring methods, we eliminate the unknowns corresponding to the interior of the subregions. Problem (6) becomes a Schur complement problem on V h (Γ ): find u Γ V h (Γ ) s.t. a Γ (H(u Γ ), H(v Γ )) = f(h(v Γ )), v Γ V h (Γ ). (7) For simplicity, we denote V h (Γ ) by V and let a(, ) =a Γ (H( ), H( )) be the inner product on V. Problem (7) can be formulated on V as follows: find u V s.t. a(u, v) =f(v), v V. (8)
75 D. Stefanica 4 A Balancing Algorithm for Mortars In this section, we introduce a new balancing algorithm for mortar finite elements. Our results can be extended to second order self-adjoint elliptic problems with mixed boundary conditions discretized by geometrically nonconforming mortars, and to three dimensional problems. We solve (8) using the technique outlined in Section. To do so, we need to introduce a coarse space V and local spaces V i, i =:N. The major difference between the classical balancing method and our algorithm for mortars is related to the extended subregions Ωi e, which replace the individual subregions in the definition of the local bilinear forms ã i(, ). An important role in the balancing algorithm is played by the counting functions associated with the interface nodes of each extended subregion. In [], we have shown that defining ã i(, ) only on Ω i does not lead to a convergent algorithm. Extended Subregions: The extended subregion eω i is defined as the union of Ω i and all its neighbors that have a mortar side opposite Ω i.let N i be the set of corner nodes of Ω i, all the nodes on the mortar sides of Ω i, and all the nodes on the mortar sides opposite the nonmortar sides of Ω i. The counting function ν i : Γ R corresponding to Ω i is a mortar function taking the following values at the genuine degrees of freedom: 8 < number of sets N j with x N j, if x N i; ν i(x) =, if x/ N i; :, if x Ω i Ω N. In the geometrically conforming case, the value of ν i at every interior node of the mortar sides where ν i does not vanish is equal to, and Range(ν i) {,,...,4}. Let ν i be the mortar function with nodal values ν i (x) =/νi(x) if νi(x) and ν i (x) = otherwise. As in the continuous finite element case, ν i form a partition of unity, i.e., ν i =. i= Coarse Space V : The coarse space V has one basis function, H(ν i ),theharmonic extension of ν i, per subregion Ωi. The bilinear form a is exact, i.e., ã(, ) = a(, ). Therefore, a(p u, H(ν i )) = a(u, H(ν i )), and a((i P )u, H(ν i )) =, u V. (9) Local Spaces: The local space V i is associated with the subregion Ω i, is embedded in V, i.e., V i V, and consists of piecewise harmonic functions which vanish at all the genuine degrees of freedom of Γ \ N i. The bilinear form ã i(, ) :V i V i R is defined using the extended subregion Ω e i : ã i(v i,w i) = X H(I h (ν iv i)) H(I h (ν iw i)) dx, () Ω j Ω j e Ω i where I h : L (Ω) V is the nodal basis interpolation onto the mortar space V. The projection-like operator T i is given by T i = I i et i,where
A Balancing Algorithm for Mortar Methods 75 ei (shaded) corresponding Fig.. All possible instances of extended subregions Ω to one subregion Ωi (center in each picture). The values of the counting function νi at the corners of Ωi are recorded. Mortar sides are marked with an additional solid line. a i (Tei u, vi ) = a(u, vi ), vi Vi. () ei contains more than one subregion, then any vi Vi ei = Ωi, i.e., if Ω If Ω ei \ Ωi. The problem () is well posed since it is a Poisson problem vanishes on Ω ei \ Ωi. ei with zero Dirichlet boundary conditions on Ω on Ω ei = Ωi, then all the sides of Ωi are mortars; cf. Figure, upper left If Ω picture. This corresponds to the case of a floating subregion in the classical balancing algorithm, and requires using balanced functions. Note that H(νi νi ) is equal to on Ωi, and therefore a i (Tei u, H(νi )) = H(Ih (νi Tei u)) H(νi νi ) dx =. Ωi For the local problem () to be solvable, u must satisfy a(u, H(νi )) =, () for every floating subregion Ωi. Such functions are called balanced functions. From (9), we conclude that any function in Range(I P ) is balanced. ei = Ωi, the local problem () corresponds to a pure Neumann Moreover, if Ω problem. We make the solution unique by requiring Tei u to satisfy H(Ih (νi Tei u)) dx =. () Ωi
75 D. Stefanica The preconditioned operator of our balancing algorithm for mortars is T bal = P +(I P )(T + + T N)(I P ). The convergence analysis of T bal relies on that of the Neumann-Neumann operator T N N = P + T + + T N,since κ(t bal ) κ(t N N ) ; cf., e.g., [8]. However, Neumann-Neumann algorithms with the spaces and approximate solvers considered in this paper would not converge. 5 Condition number estimate The condition number estimate for our algorithm is based on abstract Schwarz theory; see, e.g., []. A technical results has to be proven first, and the techniques are somewhat different for floating and non-floating regions: Lemma. Let u V and let u i = H(I h (ν i (u αi))) Vi,where αi is the weighted averages of u over Ω i, i.e., Then, α i = µ(ω i) Ω i udx. (4) a(u i,u i) C`+log(H/h) ã i(u i,u i), u i Range(T i) (5) a(u i, u i) C`+log(H/h) u H ( Ω f. i ) (6) Also, if Ω i is a floating subregion, i.e., if e Ωi = Ω i,then, ã i(u i, u i)= u H ( f Ω i ). If Ω i is a nonfloating subregion, i.e., if Ωi e Ω i, then ã i(u i, u i) C` + log(h/h) u H ( Ω f. i ) Using the results of Lemma, we can show that ã i(, ) is bounded from below by a(, ), and prove that for any function in V there exists a stable splitting into local functions; see [] for detailed proofs. Lemma. There exists a constant C, not depending on the local spaces V i,such that a(u i,u i) C`+log(H/h) ã i(u i,u i), u i Range(T i), i =:N. Lemma. Let u V and let α i be the weighted averages (4) of u over Ω i. Define u V as u = α ih(ν i ) and let ui Vi be given by ui = H(I h(ν i (u i= α i))).then u = u + and a(u,u )+ u i i= ã i(u i, u i) C`+log(H/h) a(u, u). i=
A Balancing Algorithm for Mortar Methods 75 Based on the results of Lemmas and, a bound on κ(t N N ), and therefore on κ(t bal ), can be established using the abstract Schwarz theory. Theorem. The condition number of the balancing algorithm is independent of the number of subregions and grows at most polylogarithmically with the number of nodes in each subregion, i.e., κ(t bal ) C`+log(H/h) 4, where C is a constant that does not depend on the properties of the partition. 6 Numerical Results We have tested the convergence properties of our balancing algorithm for a two dimensional problem discretized by geometrically nonconforming mortar finite elements. The model problem was the Poisson equation on the unit square Ω with mixed boundary conditions. We partitioned Ω into N = 6,, 64, and 8 geometrically nonconforming rectangles, and Q elements were used in each rectangle. For each partition, the number of nodes on each edge, H/h,wastakentobe, on average, 4, 8, 6, and, respectively, for different sets of experiments. The preconditioned conjugate gradient iteration was stopped when the residual norm had decreased by a factor of 6. The experiments were carried out in MATLAB. We report iteration counts, condition number estimates, and flop counts of our algorithm in Table. Table. Convergence results, geometrically nonconforming mortars. N H/h Iter Cond Mflops N H/h Iter Cond Mflops 6 4 9. 4.7e- 64 4 4 9.9 4.e+ 6 8.8.6e+ 64 8 5..6e+ 6 6 4..6e+ 64 6 7.4 9.4e+ 6 5..e+ 64 9.9 7.e+ 4 9.6.5e+ 8 4 4..e+ 8 4. 7.e+ 8 8 5..6e+ 6 5.9 4.5e+ 8 6 8.7.e+ 6.6.e+ 8 9.9.5e+ Our balancing algorithm has similar scalability properties as those of the classical balancing algorithm. When the number of nodes on each subregion edge, H/h, was fixed and the number of subregions, N, was increased, the iteration count showed only a slight growth. When H/h was increased, while the partition was kept unchanged, the small increase in the number of iterations was satisfactory. The condition number estimates exhibited a similar dependence, or lack thereof, on N and H/h.
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