Quasilinear Algorithm for Elastodynamic Boundary Integral Equation Method

We present a fast algorithm with small memory storage to compute the spatiotemporal boundary integral equation method (ST-BIEM) particularly for the elastodynamic problem. The time complexity of the spatiotemporal convolution and memory consumption to store the integral kernel and convolved variables are originally of $\mathcal O(N^2M)$ in ST-BIEM for a given number of discretized fault elements $N$ and time steps $M$. Such huge costs of ST-BIEM are reduced to be of $\mathcal O(N \log N)$ by our methods, called the fast domain partitioning hierarchical matrices (FDP=H-matrices). FDP=H-matrices are natural extensions of the fast domain partitioning method (FDPM) and the hierarchical matrices (H-matrices), combined with newly developed two algorithms. After developing new methods, we test the cost and accuracy of FDP=H-matrices both analytically and numerically.