Abstract
This work extends the general form of the Multiscale HybridMixed (MHM) method for the secondorder Laplace (Darcy) equation to general nonconforming polygonal meshes. The main properties of the MHM method, i.e., stability, optimal convergence, and local conservation, are proven independently of the geometry of the elements used for the first level mesh. More precisely, it is proven that piecewise polynomials of degree k and k+1, k 0, for the Lagrange multipliers (flux), along with continuous piecewise polynomial interpolations of degree k+1 posed on secondlevel submeshes are stable if the latter is fine enough with respect to the mesh for the Lagrange multiplier. We provide an explicit sufficient condition for this restriction. Also, we prove that the error converges with order k +1 and k +2 in the broken H1 and L2 norms, respectively, under usual regularity assumptions, and that such estimates also hold for nonconvex; or even nonsimply connected elements. Numerical results confirm the theoretical findings and illustrate the gain that the use of multiscale functions provides.
Original language  English 

Pages (fromto)  197237 
Number of pages  41 
Journal  Numerische Mathematik 
Volume  145 
Issue number  1 
Early online date  20 Feb 2020 
DOIs  
Publication status  Published  1 May 2020 
Keywords
 Darcy equation
 polygonal meshes
 multiscale finite element method
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Gabriel Barrenechea
Person: Academic