123 Editors

We present an adaptive mimetic finite difference method for the approximate solution of variational inequalities. The adaptive strategy is based on a heuristic hierarchical type error indicator. Numerical experiments that validate the performance of the adaptive MFD method are also presented. 1 The Obstacle Problem Throughout the paper we will use standard notations for Sobolev spaces, norms and seminorms. For a bounded domain D in R, we denote by H.D/ the standard Sobolev space of order s 0, and by k kHs.D/ and j jHs.D/ the usual Sobolev norm and seminorm, respectively. For s D 0, we write L.D/ instead of H.D/. H 0 .D/ is the subspace of H.D/ of functions with zero trace on @D. Let ̋ be an open, bounded, convex set of R, with either a polygonal or a Csmooth boundary WD @ ̋ . Let g WD Q gj , with Q g 2 H. ̋/ and we set V g WD fv 2 H. ̋/ W v D g on g. Let us introduce the bilinear form a. ; / W H. ̋/ H. ̋/ ! R defined by a.u; v/ WD R ̋ ru rv dx, and the linear functional F. / W H. ̋/ ! R with F.v/ WD R ̋ f v dx, where we assume f 2 L. ̋/. Finally, we define the function 2 H. ̋/ with g on and the convex space K WD fv 2 V g W v a.e. in ̋g: P.F. Antonietti M. Verani ( ) MOX – Modelling and Scientific Computing Dipartimento di Matematica “F. Brioschi”, Politecnico di Milano, Piazza Leonardo da Vinci 32, I-20133, Milano, Italy e-mail: [email protected]; [email protected] L. Beirão da Veiga Dipartimento di Matematica, Università di Milano, Via Saldini 50, I-20133, Milano, Italy e-mail: [email protected] A. Cangiani et al. (eds.), Numerical Mathematics and Advanced Applications 2011, DOI 10.1007/978-3-642-33134-3 1, © Springer-Verlag Berlin Heidelberg 2013 3 4 P.F. Antonietti et al. We are interested in solving the following variational inequality: 8 < : Find u 2 K such that a.u; v u/ F.v u/ 8v 2 K: (1) It is well known (see e.g. [6]) that under the above data regularity assumption, the elliptic obstacle problem (1) admits a unique solution u 2 H. ̋/. 2 The Mimetic Discretization In this section we recall the mimetic discretization for the obstacle problem (1) (see [2] for more details). Let ̋h ̋ be a polygonal approximation of ̋ , in such a way that all vertexes of ̋h which are on the boundary of ̋h are also on the boundary of ̋ . The polygonal domain ̋h represents the computational domain for the method. With a little abuse of notation, we also denote by ̋h a partition of the above introduced computational domain into polygons E . We assume that this partition is conformal, i.e., intersection of two different elements E1 and E2 is either a few mesh points, or a few mesh edges (two adjacent elements may share more than one edge) or empty. We allow ̋h to contain non-convex elements. Note moreover that, differently from conforming finite element meshes, T-junctions are now allowed in the mesh; indeed, this are included in the above conditions simply by splitting single edges into two new (aligned) edges. For each polygon E , kE denotes its number of vertexes, jEj its area, hE its diameter and h WD max E2 ̋h hE: We denote the set of mesh vertexes and edges by Nh and Eh, the set of internal vertexes and edges by N 0 h and E 0 h , the set of boundary vertexes and edges by N @ h and E @ h . The set of vertexes and edges of a particular elementE are denoted by N E h and E E h , respectively. Moreover, we denote a generic mesh vertex by v, a generic edge by e and its length both by he and jej. A fixed orientation is also set for the mesh ̋h, which is reflected by a unit normal vector ne, e 2 Eh, fixed once for all. For every polygon E and edge e 2 E E h , we define a unit normal vector nE that points outside E . The mesh is assumed to satisfy the following shape regularity properties, which have already been used in [7]. There exist An integer numberNs independent of h; A real positive number independent of h; A compatible sub-decomposition Th of every ̋h into shape-regular triangles, An Adaptive MFD Method for the Obstacle Problem 5 such that (H1) Any polygon E 2 ̋h admits a decomposition ThjE formed by less than Ns triangles; (H2) Any triangle T 2 Th is shape-regular in the sense that the ratio between the radius rT of the inscribed ball and the diameter hT of T is bounded from below by ; i.e. 0 < rT hT . The discretization of problem (1) requires to discretize a scalar field in H. ̋/. To this aim, we start introducing the degrees of freedom for the discrete approximation space. The discrete space Vh is defined as follows: a vector vh 2 Vh consists of a collection of degrees of freedom vh WD fvgv2Nh ; one per mesh vertex, e.g. to every vertex v 2 Nh, we associate a real number vv. The scalar vv represents the nodal value of the underlying discrete scalar field. The number of unknowns is equal to the number of vertexes of the mesh. We also define the discrete space V g h Vh of functions which satisfy the Dirichlet boundary condition: V g h WD fvh 2 Vh W v h D g.v/ 8v 2 N @ h g : Accordingly, V 0 h represents the space of discrete functions which vanish at the boundary nodes. We define the following interpolation operator from the spaces of smooth enough functions to the discrete space Vh. For every function v 2 C . N̋ / \ H. ̋/, we define vI 2 Vh by v I WD v.v/ 8v 2 Nh: Moreover, we analogously define the local interpolation operator from C . N E/ \ H.E/ into VhjE given by v I WD v.v/ 8v 2 N E h : We endow the space Vh with the following discrete seminorm