Domain decomposition vector schemes for second-order evolution equations (Q5928731)

Domain decomposition methods are widely used in mathematical physics problems. As a matter of fact they give possibility to use parallel calculations scheme by applying powerful computers. The second-order evolution differential equation \[ \frac {\partial ^2u}{\partial t^2}= Lu, \quad \mathbf x \in G, \quad t>0, \quad \tag{1} \] where \(L\) is a self-ajoint operator of the second order is considered. To find an approximate solution of the problem (1) with the boundary conditions \[ u(\mathbf x, t) = 0, \quad \mathbf x \in \partial G, \quad t>0, \] and the initial conditions \[ u(\mathbf x, t) = u_0(\mathbf x), \quad \mathbf x \in G, \frac{\partial u(\mathbf x, 0)}{\partial t} = u_1(\mathbf x), \quad \mathbf x \in G, \] some unconditionally stable vector difference schemes of the domain decomposition are constructed. It is turned out that the initial selfadjoint operator in a grid Hilbert space has an additive representation with an arbitrary number of operators. A transfer to vector schemes i.e. instead of one grid function a vector grid function is introduced, is used. Stability and convergence questions are analyzed on a base of general results from the theory of difference schemes.