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Two non-overlapping domain decomposition methods are presented for the mixed finite element formulation of linear elasticity with weakly enforced stress symmetry. The methods utilize either displacement or normal stress Lagrange multiplier to impose interface continuity of normal stress or displacement, respectively. By eliminating the interior subdomain variables, the global problem is reduced to an interface problem, which is then solved by an iterative procedure. The condition number of the resulting algebraic interface problem is analyzed for both methods. A multiscale mortar mixed finite element method for the problem of interest on non-matching multiblock grids is also studied. It uses a coarse scale mortar finite element space on the non-matching interfaces to approximate the trace of the displacement and impose weakly the continuity of normal stress. A priori error analysis is performed. It is shown that, with appropriate choice of the mortar space, optimal convergence on the fine scale is obtained for the stress, displacement, and rotation, as well as some superconvergence for the displacement. Computational results are presented in confirmation of the theory of all proposed methods.
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An efficient flux‐variable approximation scheme for Darcy's flow
Abstract We present an efficient numerical method to approximate the flux variable for the Darcy flow model. An important feature of our new method is that the approximate solution for the flux variable is obtained without approximating the pressure at all. To accomplish this, we introduce a user‐defined parameter delta, which is typically chosen to be small so that it minimizes the negative effect resulting from the absence of the pressure, such as inaccuracy in both the flux approximation and the mass conservation. The resulting algebraic system is of significantly smaller degrees of freedom, compared to the one from the mixed finite element methods or least‐squares methods. We also interpret the proposed method as a single step iterate of the augmented Lagrangian Uzawa applied to solve the mixed finite element in a special setting. Lastly, the pressure recovery from the flux variable is discussed and an optimal‐order error estimate for the method is obtained. Several examples are provided to verify the proposed theory and algorithm, some of which are from more realistic models such as SPE10.
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- PAR ID:
- 10535958
- Publisher / Repository:
- Numerical methods for Partial Differential Equations
- Date Published:
- Journal Name:
- Numerical Methods for Partial Differential Equations
- ISSN:
- 0749-159X
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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- Free Publicly Accessible Full Text
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Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.1002/num.23120
