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$$hp$$ submeshing via non-conforming finite element methods. (English) Zbl 0971.65101
The mixed boundary value problem in a bounded polygonal domain $$\Omega$$ in $${\mathbb R}^2$$ is considered for the Poisson equation. The authors give a review of grid methods with discontinuous approximations (mortar element methods) and pay special attention to the analysis of two variants of non-conforming finite element methods with piecewise polynomial approximations on triangular cells and different ways to approximate the continuity of the function on the cutting line (interface). Estimates of convergence are suggested which depend on the grid parameter and degree of used polynomials.
The authors present numerical results in details and conclude the paper by a desription of possible generalization for three-dimensional domains. There are remarks about optimality of the methods “what is meant that the method should perform as well as the conforming method”, but it is clear that it can not take the place since approximations do not be belong to the original energy space. Besides, the correctness of approximating problems is rather weak since the constant in the inf-sup condition (3.2) might tend to 0.

##### MSC:
 65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs 65N12 Stability and convergence of numerical methods for boundary value problems involving PDEs 35J25 Boundary value problems for second-order elliptic equations 65N50 Mesh generation, refinement, and adaptive methods for boundary value problems involving PDEs
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