Inhomogeneous Dirichlet Boundary Conditions Research Articles

A finite‐difference, time‐domain solution for three‐dimensional electromagnetic modeling

We have developed a finite‐difference solution for three‐dimensional (3-D) transient electromagnetic problems. The solution steps Maxwell’s equations in time using a staggered‐grid technique. The time‐stepping uses a modified version of the Du Fort‐Frankel method which is explicit and always stable. Both conductivity and magnetic permeability can be functions of space, and the model geometry can be arbitrarily complicated. The solution provides both electric and magnetic field responses throughout the earth. Because it solves the coupled, first‐order Maxwell’s equations, the solution avoids approximating spatial derivatives of physical properties, and thus overcomes many related numerical difficulties. Moreover, since the divergence‐free condition for the magnetic field is incorporated explicitly, the solution provides accurate results for the magnetic field at late times. An inhomogeneous Dirichlet boundary condition is imposed at the surface of the earth, while a homogeneous Dirichlet condition is employed along the subsurface boundaries. Numerical dispersion is alleviated by using an adaptive algorithm that uses a fourth‐order difference method at early times and a second‐order method at other times. Numerical checks against analytical, integral‐equation, and spectral differential‐difference solutions show that the solution provides accurate results. Execution time for a typical model is about 3.5 hours on an IBM 3090/600S computer for computing the field to 10 ms. That model contains [Formula: see text] grid points representing about three million unknowns and possesses one vertical plane of symmetry, with the smallest grid spacing at 10 m and the highest resistivity at 100 Ω ⋅ m. The execution time indicates that the solution is computer intensive, but it is valuable in providing much‐needed insight about TEM responses in complicated 3-D situations.

Stabilization techniques for spectral methods

We introduce a stabilized treatment of spectral methods. The condition number of the spectral systems is highly improved. Elliptic and biharmonic problems are considered. Suitable interpolants in the case of inhomogeneous Dirichlet boundary conditions are presented. For a direct solver the improvements with respect to rounding error propagation are numerically demonstrated.

On the convergence rate of the boundary penalty method

AbstractThe convergence rate of the boundary penalty finite element method is discussed for the Poisson equation with inhomogeneous Dirichlet boundary conditions on a polygonal domain. It is proved that, under certain circumstances, an optimal convergence rate may be achieved which agrees with the rate obtained recently in the numerical experiments by Utku and Carey.

The Penalty Correction Method for Elliptic Boundary Value Problems

A new finite element method is introduced for approximating elliptic boundary value problems with inhomogeneous Dirichlet boundary conditions. This new method, the penalty correction method, leads to approximations which are “optimal” in $H^1 ,L^2 $, and negative Sobolev spaces without very stringent assumptions on the approximating subspaces, viz., the subspaces need not satisfy the Dirichlet boundary conditions or inverse properties. Computationally, the method involves solving the same matrix system, corresponding to Babuska’s penalty method, for a number of input vectors. Thus one can take full advantage of matrix decomposition algorithms available. The penalty correction method also gives rise to high order approximations to the normal derivative of the solution at the boundary without computing normal derivatives.