A GPU-based multi-level algorithm for boundary value problems

A novel geometric single-grid multi-level (GSGML) algorithm is presented for the numerical solution of boundary value problems (BVPs). The algorithm is specifically designed for parallel scalability with high occupancy of streaming processors inside many-core architectures like the Graphics Processing Unit (GPU). The algorithm consists of iterative, superposed operations on a single grid, and is composed of two full-grid routines: a restriction and a coarsened interpolation-relaxation. The restriction is a local average of scalar or vector fields, and the interpolation-relaxation is a simultaneous application of coarsened finite-difference operators and interpolations in complementary subsets of the grid. The routines are scheduled in a saw-like refining cycle. Convergence to machine precision is achieved repeating the full cycle using residuals and superposition. The algorithm is attractive for its computational simplicity and provides a competitive GPU-based fast solver for elliptic partial differential equations like the Poisson-Helmholtz equation. Applications shown in this work include the deformation of two-dimensional grids, the computation of three-dimensional streamlines for a singular trifoil-knot vortex and the calculation of three-dimensional electric potentials in heterogeneous dielectric media. (C) 2019 Elsevier B.V. All rights reserved.