Numerical solution of differential and integral equations
We propose and prove the minimum principles of quadratic energy functionals for two boundary value problems. The normal or tangential components of the vector field are given at the boundary in these problems. The scalar potential and three cartesian components of the vector potential are the new unknown functions. Each of them satisfy Poisson equation inside the domain. The first boundary value problem corresponds to zero tangential components of the vector potential at the boundary and zero average value of the scalar potential. The scalar potential and the normal component of the vector potential equal zero at the boundary in the second problem. These are the main boundary conditions for minimization of the fuctionals.
The problem for the scalar potential is separate in both cases. The rest functionals differ from three Dirichlet ones for each component of the vector potential only by the main boundary conditions, which involve all components, if the domain is not a parallelepiped. These conditions are trivial in comparison with usual usage of the vector potential only, when it is necessary to design previously a boundary vector function with given curl.
We use piece-wise linear functions to approximate each unknown function. If the domain is a polyhedron, the positive definition of the matrix of the finite element method follows immediately from the proved principle. Otherwise it ought be proved independently, since the main boundary condition can not be fulfilled point by point at a curved surface. It is done in the particular case, when the domain is a circular cylinder.
The method is applied to calculate magnetic field in the Earth's magnetosphere.
Note. Abstracts are published in author's edition
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