Bibliography

(eng): Many problems in science and engineering have their mathematical formulation which leads to solving an operator equation Au = f , u ∈ M , f ∈ H , (1) where H is a Hilbert (or Banach) space, M is a subspace of H, u is a solution of (1) and A is an operator on M. In particular, we will focus on differential operators. There are a lot of methods for solving (1) and one of them is the so called variational approach which is based on finding the minimum of corresponding energy functional. In our text we represent the variational principle for solving some particular problems using a finite elements method (FEM) for discretization of energy functionals. Minimization procedures of energy functionals require the knowledge of a gradient. If an exact gradient form is not available or difficult to compute, a numerical approximation can be assembled locally. The key feature is the sparsity of Hessian matrix which significantly affects the time and memory demands of evaluations