A reliable and computable a posteriori error bound is derived for the mixed Ciarlet–Raviart method for the equation ΔΔ u+κ^2u=f(x) , x∈Ω⊂ R^2 , with the first boundary condition and a piecewise constant κ≥ 0 . Several authors derived residual type a posteriori error bounds at assumptions that κ≡ 0 and the domain is polygonal, none of which is used in the paper. In case of a piecewise smooth boundary, we consider the mixed method with the triangular Lagrange finite elements of the 3^d order, which, in general, are curvilinear along the boundary. This provides an approximation to the boundary, matching the finite elements in accuracy. Our bounds belong to the class of a posteriori functional majorants and are evaluated with help of functions from the testing C^1 space. By the reasons of accuracy and simplicity, this space is generated by the finite elements with the domains, coinciding with the domains of the Lagrange elements, and singular rational coordinate functions.

We present guaranteed, robust and computable a posteriori error bounds for approximate solutions of the equation ΔΔu + κ2 u = f by classical and mixed Ciarlet-Raviart finite element methods. We concentrate on the case when the reaction coefficient κ is subdomain (finite element) wise constant and chaotically varies between subdomains in the sufficiently wide range. It is proved that the bounds for the classical FEM's are robust with respect to κ ∊ [0,ch−2], where c = const and h is the maximal size of finite elements, and possess additional useful features. The coefficients in fronts of two typical norms in their right parts only insignificantly worse than those for κ ≡ const, and the bounds can be calculated without resorting to the equilibration procedures. Besides, they are sharp at least for low order methods, if the testing moments and deflection in their right parts are found by accurate recovery procedures. The technique of derivation of the bounds is based on the approach similar to one used in our preceding papers for simpler problems.

For the efficient error control of numerical solutions of the solid mechanics problems, the two requirements are important: an a posteriori error bound has sufficient accuracy and computation of the bound is cheap in respect to the arithmetic work. The first requirement can be formulated in a more specific form of consistency of an a posteriori bound, assuming that it is not improvable in the order and, at least, coincides in the order with the a priori error estimate. Several new a posteriori error bounds are presented, which improve accuracy and reduce the computational cost. Also for the first time a new consistent guaranteed a posteriori error bound is suggested.

In DD (domain decomposition) methods, the main contribution to the computational work is due to the two major components — solvers for local Dirichlet problems on subdomains of decomposition and local problems on their faces. Without loss of generality, we assume that the domains of FE's (finite elements) serve as subdomains of decomposition. At that, under the conditions of shape regularity, optimization of these components is reduced to obtaining fast preconditioners-solvers for the stiffness matrix of the p reference element and the Schur complement, related to its boundary.

In this paper, we advocate the ”classical” approach to the a posteriori error estimation, which for the theory elasticity problems stems from the Lagrange and Castigliano variational principles. In it, the energy of the error of an approximate solution, satisfying geometrical restrictions, is estimated by the energy of the difference of the stress tensor corresponding to the approximate solution and any stress tensor, satisfying the equations of equilibrium. Notwithstanding a popular point of view that the construction of equilibrated stress fields requires considerable computational effort, we show that it can be practically always done for a number of arithmetic operations, which is asymptotically optimal. Numerical experiments show that a posteriori error estimators, based on the use of exactly equilibrated stress fields, provide very good coefficients of effectiveness, which in many cases can be convergent to the unity. At the same time they have linear complexity and are robust.

An important for applications, the class of hp discretizations of second-order elliptic equations consists of discretizations based on spectral finite elements. The development of fast domain decomposition algorithms for them was restrained by the absence of fast solvers for the basic components of the method, i.e., for local interior problems on decomposition subdomains and their faces. Recently, the authors have established that such solvers can be designed using special factorized preconditioners. In turn, factorized preconditioners are constructed using an important analogy between the stiffness matrices of spectral and hierarchical basis hp -elements (coordinate functions of the latter are defined as tensor products of integrated Legendre polynomials). Due to this analogy, for matrices of spectral elements, fast solvers can be developed that are similar to those for matrices of hierarchical elements. Based on these facts and previous results on the preconditioning of other components, fast domain decomposition algorithms for spectral discretizations are obtained.

Preconditioners for the internal sriffness matrices of the spectral and hierarchical (with the coordinate polynomials produced by the tensor products of the integrated Legendre’s polynomials) reference elements are considered. It is shown that fast preconditioners-solvers for the reference elements of one type may be easily adapted to the other type with the same in the order arithmetical complexity. This immediately widens the number of fast preconditioners-solvers for the both types of elements.

Highly parallelizable domain decomposition Dirichlet-Dirichlet solvers for hp-version finite element methods on angular quasiuniform triangular meshes are studied under different assumptions on a reference element. The edge coordinate functions of a reference element are allowed to be either nodal with special choices of nodes, or hierarchical polynomials of several types. These coordinate functions are defined within elements as being arbitrary or discrete quasi-harmonic coordinate functions. The latter are obtained from explicit and inexpensive prolongation operators. In all situations, we are able to suggest preconditioners which are spectrally equivalent to the global stiffness matrix, which only require element-by-element and edge-by-edge operations, and which reduce computational cost. In this way, elimination is avoided when dealing with the interface problem. The domain decomposition algorithms essentially use prolongation operators from the interface boundary inside the subdomains of the decomposition according to the approach initially used for the hp-version finite element methods with quadrilateral elements [S.A. Ivanov, V.G. Korneev, Izv. Vyssh. Uchebn. Zaved. 395 (1995) 62-81; Technische Universität Chemnitz-Zwickau, Preprint SPC 95-35, 1995, 1-15, and Preprint SPC 95-36, 1995, 1-14; Math. Modeling 8 (1996) 63-73].