Keywords: coupled heat and moisture transfer, Künzel model, Schur complement method, parallel computing, domain decomposition.

problems are still more popular in many engineering branches. The popularity stems from the effort to solve very complicated tasks and to obtain very good agreement with reality or experiments. The coupled heat and moisture transfer plays an important role in civil engineering and it is often connected with mechanical analysis.

The coupled heat and moisture transfer is based on the Künzel model published in his thesis in 1995. Basic equations are summarized and discretization based on the finite element method is shown. Definitions of the capacity and conductivity matrices are given. Time discretization is done using the generalized trapezoidal method. The final equations are non-linear non-symmetric algebraic equations. The non-linearity is solved by the Newton method.

Many methods for solution of systems of equations are excluded in the case of non-symmetric systems. In this contribution, the Schur complement method is used for solution of large systems. The Schur complement method is primal domain decomposition method, i.e. the original unknowns are used during the whole computation. The method is also non-overlapping, i.e. the subdomains do not overlap each other. Each finite element belongs to only one subdomain. The Schur complement method is in fact a suitably written Gaussian elimination method. The local subdomain matrices are factorised and the internal unknowns which are shared by only one subdomain are eliminated. It means that only interface unknowns remain in the problem. The system containing only the interface unknowns is called the reduced system. It can be solved by direct as well as iterative methods. If the Schur complement method is applied to a symmetric positive definite problem, the Cholesky factorisation can be used. In the case of non-symmetric problems, the LU factorisation can be used for elimination of the internal unknowns and for solution of the reduced system.

Application of the Schur complement method to solution of coupled heat and moisture transfer is documented on an example of analysis of a building brick. With respect to the relatively complicated shape, a finite element mesh with 257,774 unknowns is used. The mesh is split to four and eight submeshes and the computation is performed on one, four and eight processors. Very good speedup is obtained.

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