Finite-Difference Schemes

Due to its relative simplicity, Finite Difference (FD) analysis was historically the first numerical technique for boundary value problems in mathematical physics. The excellent review paper by V. Thomée [Tho01] traces the origin of FD to a 1928 paper by R. Courant, K. Friedrichs and H. Lewy, and to a 1930 paper by S. Gerschgorin. However, the Finite Element Method (FEM) that emerged in the 1960s proved to be substantially more powerful and flexible than FD. The modern techniques of hp-adaption, parallel multilevel preconditioning, domain decomposition have made FEM ever more powerful (Chapter 3). Nevertheless, FD remains a very valuable tool, especially for problems with relatively simple geometry.