# Numerical Method for Equation System of two depending Equation Systems

I am searching a solution method for the following equation system of equation systems:

Let $$A \in \mathbb{R}^{n \times n}$$ be an invertible Matrix, $$f, b_1, b_2 \in\mathbb{R}^n$$ given vectors and $$C_i \in \mathbb{R}$$ constants. I am searching for a numerical method to find the solution vectors $$x_1, x_2 \in\mathbb{R}^n$$ such that

\begin{align} C_1A x_1 + C_2A x_2 &= C_1b_1 \\ C_3Ax_1 + C_4 Ax_2 &= f + C_3b_1 - C_5 b_2 \end{align}

I know that I also can write this as a equation system i.e. $$$$\begin{pmatrix} C_1 A & C_2 A \\ C_3 A & C_4 A \end{pmatrix} \begin{pmatrix} x_1 \\ x_2 \end{pmatrix} = \begin{pmatrix} 0 \\ f \end{pmatrix} + \begin{pmatrix} C_1 & 0\\ C_3 & -C_5 \end{pmatrix}\begin{pmatrix} b_1 \\ b_2 \end{pmatrix}$$$$

I am happy and looking forward getting some suggestions.

• If you combine the right hand side of your second equation. You have your system in a general form: Ax=b. You may then make use of a linear algebra package of your choice to solve the system (matlab / LAPACK etc). Do you have performace problems? Are there any properties of A you could make use of? – MPIchael Sep 25 '19 at 14:24
• Hi @MPIchael, A is a stiffness matrix resulting from an elliptic BVP i.e. A = BCB^T where B are the derivatives of the ansatz functions and C is Youngs Modulus. Of course b_i is a vector of the form B*f_i – user29088 Sep 25 '19 at 16:35
• That means that A is sparse, right? In that case any iterative solver should be quite efficient to solve it. – MPIchael Sep 25 '19 at 18:38