Abstract
Solutions \(u \in H_0^1 (\Omega ) \cap H^2 (\Omega )\) of a semilinear elliptic boundary value problem, \(Au + f(x,u,\lambda ) = 0\) (with \(f_u (x,u,\lambda )\) bounded below) can be put into a one-to-one correspondence with zeros \(c \in \mathbb{R}^d \) of a function \(c \to B(c,\lambda ) \in \mathbb{R}^d \). Often d is small. The function \(B(c,\lambda )\) is called the bifurcation function. It can also be shown that the eigenvalues of the matrix \(B_c (c,\lambda )\) characterize the stability properties of the solutions of the elliptic problem as rest points of \(u_t + Au + f(x,u,\lambda ) = 0\). A finite element method that can be used for computing B and B c has recently been proposed. An overview of these results and the finite element method is given.
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Smiley, M.W. Numerical bifurcation and stability analysis for steady-states of reaction diffusion equations. Numerical Algorithms 14, 211–225 (1997). https://doi.org/10.1023/A:1019117130906
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DOI: https://doi.org/10.1023/A:1019117130906