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Solving the Signorini Problem on the Basis of Domain Decomposition

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Solving the Signorini Problem on the Basis of Domain Decomposition Solving the Signorini Problem on the Basis of Domain Decomp osition Techniques J Sch ob erl Linz Abstract Solving the Signorini Problem on the Basis of Domain Decomp osition Techniques The nite element discretization of the Signorini Problem leads to a large scale constrained minimization problem To improve the convergence rate of the pro jection metho d preconditioning must b e develop ed To b e eective the relative condition numb er of the system matrix with resp ect to the preconditioning matrix has to b e small and the applications of the preconditioner as well as the pro jection onto the set of feasible elements have to b e fast computable In this pap er we show how to construct and analyze such preconditioners on the basis of domain decomp osition techniques The numerical results obtained for the Signorini problem as well as for plane elasticity problems conrm the theoretical analysis quite well AMS Subject Classications T J N F K Key words contact problem variational inequality domain decomp osition precon ditioning Zusammenfassung o sung des Signorini Problems auf der Basis von Gebietszerlegungs Die Au metho den Die Finite Elemente Diskretisierung des SignoriniProblems fuhrt zu einem restringierten Minimierungsproblem mit vielen Freiheitsgraden Zur Verb es serung der Konvergenzrate des Pro jektionsverfahren mussen Vorkonditionierungs techniken entwickelt werden Um ezient zu sein mu die relative Konditionszahl der Systemmatrix in Bezug auf die Vorkonditionierungsmatrix klein sein und die assige Menge Anwendungen der Vorkonditionierung als auch der Pro jektion in die zul mussen schnell b erechenbar sein In dieser Arb eit wird die Konstruktion und Analyse solcher Vorkonditionierer auf Gebietszerlegungsbasis dargestellt Numerische Ergeb at nisse fur das Signorini Problem und Kontaktprobleme aus der eb enen Elastizit stimmen mit der theoretischen Analyse gut ub erein orderung der This research has b een supp orted by the Austrian Science Foundation Fonds zur F wissenschaftlichen Forschung FWF under pro ject grant PTEC Intro duction The contact problem is an imp ortant problem in computational mechanics An elastic b o dy is deformed due to volume and surface forces but the b o dy should not p enetrate a given rigid obstacle This leads to unilateral b oundary conditions called contact conditions We refer to for mathematical mo deling and analysis In this pap er we are interested in fast numerical algorithms for solving the nite dimen sional constrained minimization problem arising from the nite element discretization of the Signorini problem or elastic contact problem There are clas sical iterative metho ds like p oint pro jection metho ds and p oint overrelaxation metho ds These metho ds suer from slow convergence rates on ne meshes Multigrid metho ds have b een successfully applied to obstacle problems with inequality constraints in the whole domain by Domain decomp osition metho ds for variational inequalities have b een investigated in But these domain decomp osition metho ds dif fer from our metho d based on domain decomp osition preconditioners Boundary element metho ds have also b een applied for contact problems in For the sake of simplicity we will consider the Signorini Problem for the Poisson equa tion to develop and analyze the algorithms but we also provide numerical results for the contact problem in elasticity For the Signorini problem the unknown function u is restricted from b elow on the Signorini part of the b oundary In the classical form the Signorini problem C reads as follows u f in u on D u on N n u u u g u g on C n n d The domain is supp osed to b e b ounded in R d or with a Lipschitzcontinuous b oundary such that meas and meas D N C D C The nite element discretization of the weak form of leads to the nite dimensional Constrained Minimization Problem CMP T T v Av f v Find u K J u inf J v with J v v K or to the equivalent variational inequality T T Find u K u Av u f v u v K The mesh size is denoted by h the total numb er of unknowns is N the numb er of unknowns on is N The solution u the symmetric and p ositive denite sp d system matrix A C C and the righthand side vector f are split into contact b oundary comp onents C and inner plus Neumann b oundary comp onents I ie u A A f C C CI C u A f u A A f I IC I I N The convex set of feasible functions K V R is dened by K fv V v g g C C where v g is meant comp onentwise C C To solve the nite dimensional constrained minimization problem the pro jection metho d is applied This metho d is not very p opular in practice b ecause its convergence rate is very slow on ne meshes unless go o d preconditioning is applied The goal of this pap er is to present preconditioning techniques satisfying the following prop erties The relative condition numb er of A with resp ect to C is small the op eration C v is fast executable the pro jection P with resp ect to the C energy norm onto K is fast computable with the sp d preconditioning matrix C Multilevel preconditioners as well as domain decomp osition DD preconditioners have b een develop ed for satisfying the rst two re quirements quite well The basic concept of Dirichlet DD is the approximative decoupling of the global FE space into inner and extended b oundary subspaces The pro jection in volves only the b oundary subspace To implement this pro jection the dual problem is intro duced This enables us to use wellknown b oundary preconditioners To avoid two cascaded iterations until convergence we analyze also the truncated version by means of the BramblePasciak transformation intro duced in If the underlying comp onents are optimal and a uniformly rened mesh is used then the complexity is of optimal order for the two as well as three dimensional case We will combine our algorithms also with nested and adaptive concepts and study the b ehavior by means of numerical examples The rest of the pap er is organized as follows The pro jection algorithm and its approxi mative version are presented in Section Convergence in the energy norm is proved The concepts of additive DD preconditioning is shortly rep eated in Section In Section b oth concepts are combined to develop an optimal preconditioner for the pro jection algorithm The truncated variant is analyzed in Section Finally numerical results are presented in Section The Pro jection Metho d The nite dimensional CMP can b e solved by the pro jection metho d which reads as Algorithm Pro jection Metho d Cho ose an arbitrary initial guess u K For k do k k k f Au u P u C In Algorithm C is the symmetric and p ositive denite sp d preconditioning matrix We assume the sp ectral equivalence inequalities C A C in the sense of Euclidean inner pro duct with the p ositive sp ectral equivalence constants and In general these b ounds dep end on the mesh parameter h If they are indep endent of h and the op eration C v is fast executable ie via O N arithmetical op erations then we call the preconditioner C asymptotically optimal Asymptotically optimal pre conditioners can b e constructed by multilevel techniques as well as by multilevel DD techniques see Section We assume that the real p ositive relaxation parameter is chosen such that The op erator P is the pro jection onto K with resp ect to the energy norm induced by the sp d preconditioning matrix C P w V P w K kP w w k kv w k v K C C k It is straightforward to show that u converges to its limit u in C energy norm with convergence factor see eg A convergence rate estimation in Aenergy norm is necessary for the comp osite algorithm in Section However it was not available from k the literature It could not b e shown that u converges monotonically to u in Anorm but the main theorem of this section provides a monotone decay of the quadratic functional J and it estimates the convergence rate For our application the exact pro jection P is to o exp ensive to compute and so it is replaced by an approximative pro jection P This leads us to the approximative pro jection metho d Algorithm Approximative Pro jection Metho d Cho ose an arbitrary u K For k do k k k f Au u u C k k u P u The following theorem estimates the convergence rate of the approximative pro jection metho d which also includes the exact pro jection metho d with P Theorem Energy convergence rate estimate k Let u b e the sequence generated by Algorithm The relaxation parameter is chosen in the interval The approximative pro jection P fullls k k k k k k kP u u k ku u k kP u u k P P C C C with Then the estimate P k k J u J u J u holds for every k N with the convergence rate P The error in Aenergy norm is b ounded by k k J u J u ku u k A Proof First we reduce the approximative pro jection to the case of an exact pro jection k We use C A the denition of u and to obtain k k k k J u J u ku u k 1 C A k k k k k ku u k ku u k J u C C k k k k k k k ku u k kP u u k ku u k J u P P C C C k k k k k k kP u u k ku u k J u J u P P C C Due to the convexity of K there holds k k k k kP u u k kP k u u k C C uu k k with the pro jection P k u onto the straight line u u This reduces the problem to uu k the plane spanned by u u u From the sketch b elow the next estimates are obvious k u ~k k P[u, u ](u )
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