The application of the finite element method for solving linear elliptic problems requires the solution of \[ Au=b, \tag{1} \] where \(A\) is a symmetric matrix of order \(N\), and \(u\) and \(b\) are the vectors of unknown and independent term. Direct, iterative, and multigrid algorithms are commonly used to solve (1). Multigrid methods use several meshes for solving (1). Computational elements like nested iterations, coarse grid correction, transfer operators, and relaxation schemes are applied. However, engineering problems have complex geometries, which sometimes makes it difficult to generate a sequence of nested meshes. Thus, using nonnested approximation spaces is an interesting option.
This article presents an application of nonnested and unstructured multigrid methods to linear elasticity problems. A variational formulation for transfer operators and some multigrid strategies, including adaptive algorithms, are presented. Expressions for the performance evaluation of multigrid strategies and its comparison with direct and preconditioned conjugate gradient algorithms are also presented. A C++ implementation of the multigrid algorithms and the quadtree and octree data structures for transfer operators are discussed. Some two- and three-dimensional elasticity examples are analyzed.