Fast Boundary-Domain Integral Method with the H ²-matrix formulation for large scale numerical investigations

In engineering, several physical models result in inhomogeneous partial differential equations. A prototype of such an equation is the modified Helmholtz equation or also called Yukawa equation. It may result from fluid mechanics (false transient approach) or heat transfer if a semi-discretisation in time with a finite difference schema is applied. Using the Boundary Element Method for the numerical solution of such problems requires to solve a boundary-domain integral equation. The main drawback of all boundary element methods is the quadratic complexity, which exists as well for boundary-domain element methods.Here, a fast approach based on the $H^2$ concept is proposed. The focus is on the discretisation of the domain integral. Respective cluster trees for the domain and the boundary nodes are established. The integral kernels in admissible blocks are approximated with Lagrange interpolation. Further, a recompression is applied, which is here performed with a fully pivoted adaptive cross approximation. The numerical results show that the memory used to store the approximated matrices is logarithmic linear. Considering the matrix formulation of the integral kernel approximation one can reduce the storing space needed in memory to linear complexity.-concept is proposed. The focus is on the discretisation of the domain integral. Respective cluster trees for the domain and the boundary nodes are established. The integral kernels in admissible blocks are approximated with Lagrange interpolation. Further, a recompression is applied, which is here performed with a fully pivoted adaptive cross approximation. The numerical results show that the memory used to store the approximated matrices is logarithmic linear. Considering the matrix formulation of the integral kernel approximation one can reduce the storing space needed in memory to linear complexity.