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order elliptic partial differential equation, H corresponds to the finite-difference approximation in the x-direction and V corresponds to derivatives in the y-direction. The matrices H and V may include terms added to the diagonal to increase stability and accelerate convergence. Under suitable per-mutations, both H and V are tri.diagonal.
where is the inverse matrix to . Unfortunately, the most efficient general purpose algorithm for inverting an matrix--namely, Gauss-Jordan elimination with partial pivoting--requires arithmetic operations. It is fairly clear that this is a disastrous scaling for finite-difference solutions of Poisson's equation.
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The FDTD method is a discrete approximation of James Clerk Maxwell's equations that numerically and simultaneously solve in both time and 3-dimensional space.
In this paper a symmetric compact finite difference method is presented for solving nonlinear two order two point boundary scheme of the form y″ = f(t, y) with boundary conditions y(a) = A, y(b) = B. The corresponding finite difference scheme with tridiagonal matrix is given by replacing the exponent terms by Padé approximation.
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For 1D finite difference, the resulting linear system is tri-diagonal and can be solved in O(n) using the Thomas algorithm. I am trying to solve a finite difference system in 3D. Numerical Methods by Quarteroni et al goes over some theory behind extending the Thomas algorithm to block linear systems.
If the finite difference approximation is to give a tridiagonal matrix having the Sturm property then there must be restrictions on the coefficients in the differ- ential equation and on the form of the boundary conditions. The type of problem we consider is defined in Section 2.