ABSTRACT
The finite difference method and finite element method are popular numerical methods to discretize problems involving ordinary and partial differential equations. In this work, we use finite differences to approximate time derivatives, and use finite elements for spatial discretizations of variational inequalities. Basic notions related to finite difference and finite element discretizations are reviewed in the first section of this chapter. In the remaining five sections we present several bounds related to the discretization of the displacements and velocities, the evolution equation for the bonding and damage field, as well as the viscoelastic and viscoplastic constitutive laws, respectively. These bounds will be used repeatedly in later chapters. More details on theoretical analysis of the finite difference method presented in this chapter can be found in several books on the topic, e.g., [84, 115]; see also the in-depth survey articles [83, 122]. Standard references on mathematical analysis of the finite element method include [8, 11, 17, 18, 30, 54, 63,66, 93, 101,114,120].
