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Direct methods are constructed for solving quadratic programming problems with the tridiagonal Jacoby M-matrix and simple constraints. These methods are employed in block relaxation and splitting algorithms for solving two-dimensional mesh obstacle problems. The convergence of the algorithms is investigated and the results of numerical experiments are presented.We consider a difference approximation of the obstacle problem min<-|Vw| 2 dx-/udx>, X = {u^W ( 2\^)\u(x)^Q 9 xeQ}.(0.1)As a result, we arrive at a quadratic programming problem with a sparse symmetric positive definite matrix. To solve this problem in a one-dimensional case where the matrix is tridiagonal, we construct a direct method which requires O(N) arithmetic operations with N unknowns, i.e. an asymptotically (in N) optimal method. In a multidimensional case, we employ block relaxation and splitting methods based on successive solution of one-dimensional mesh problems. The quadratic programming problem minU(Aj^-(/,jok K = {yeR N |y,>0, i= 1,...,N} (0.2) y*K t 2 3 with the symmetric positive matrix A can be solved by well-known methods. Among them we can single out conjugate gradient methods developed for the sparse matrix [11] as well. In fact, these methods consist in determining a set of indices CD' = {i \ y t > 0 } at which the equalities (Ay -f\ = 0 hold. The conjugate gradient method is used to solve at each step a subsystem of the simultaneous equations Ay =/. Conjugate gradient methods converge in a finite number of iterations only due to the finite number of subspaces in the space R N but efficient estimates of the number of iterations, required for attaining a prescribed accuracy, are unknown. Methods are also available ([2,10] etc.) for solving a complementarity system [equivalent to (0.2)] tt^O, (Ay-f^O, 3^3;-A = 0, i=l,...,N (0.2') which exploit the fact that the matrix A belongs to classes (P) and (Z)[4], i.e. A is an M-matrix [16]. Unlike the conjugate gradient methods, these methods provide for a special construction of approximations to the set ω', with the exact solution y to problem (0.2) found at a step q < N.

Summary. In this paper we present new numerical approach to solve the continuous casting problem. The main tool is to use IPEC method and DDM similar to Lapin and Pieska [2002] with multilevel domain decomposition. On the subdomains we use multidecomposition of the subdomains. The IPEC is used both in the whole calculation domain and inside the subdomains. The calculation algorithm is presented and numerically tested. Several conclusions are made and discussed.

A mixed hybrid finite element method has been applied to a variational inequality with a potential second-order quasi-linear differential operator. The Lagrange multiplier method for a dual problem has been used to construct this finite element scheme. The existence and uniqueness of a solution for the resulting finite- dimensional problem has been proved, the solution iterative methods are discussed. The non-overlapping domain decomposition method combined with the mixed hybrid finite element approximation is analyzed.

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