The domain decomposition techniques for the finite element probability computational methods (Q2721929)

If the Dirichlet problem for an elliptic partial differential equation is discretized in such a way that its coefficient matrix has all its non-zero off-diagonal entries less than zero, and all row sums equal to zero, then the coefficient matrix can be regarded as a transition probability matrix. This matrix can be used to define random variables whose expected values recover the solution to the original Dirichlet problem, and a numerical method based on computing the expected values is here termed ``the probability computational method''. NEWLINENEWLINENEWLINEThe author notes that the time required to effect the probability computational method depends on the distance of interior points to the boundary. Thus, domain decomposition methods promise the possibility of acceleration as well as implementation on parallel computers. A numerical example is presented.