We consider a procedure for solving boundary value problems for elliptic homogeneous equations, known as the fundamental solutions method. We prove its applicability for some second order operators as well as for fourth order ones. The boundary conditions of an elliptic problem are approximated by using fundamental solutions of the corresponding operator with singularities located outside the domain of interest. In a specific case of the Laplacian an estimate shows that the method discussed possesses convergence properties as good as those of any method using harmonic polynomials as trial functions. Numerical examples are given, among which are simple Signorini's problems for harmonic and Lam6 operators.
Abstract. We discuss an algorithm for the numerical solution of the Obstacle Problem in which the coincidence set is considered as the prime unknown. Domain functionals are defined for which the coincidence set serves as the minimizing element. Their gradients are computed (in the sense of the material derivative), and the gradient descent method employed to minimize these functionals. Numerical example is given.
We formulate the roller contact problem in terms of variational inequality with certain pseudodifferentional operator, and discuss a finite element technique for its numerical solution.
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