Convergent and spurious solutions of nonlinear elliptic equations
Thomas Murdoch, Chris J. Budd · IMA Journal of Numerical Analysis · 1992
In this paper we investigate finite element approximations of nonlinear elliptic equations in three dimensions. By applying and extending the results of Lopez-Marcos and Sanz-Serna, we prove that the finite element approximation on a mesh of size h, has a solution Uk which converges to an exact solution of the differential equation as h→0. This solution is unique within a suitably defined stability ball Bh. For the particular nonlinear equation Δu + γ(u + up) we show that the size of Bh depends upon h only if p > 5 when it tends to zero as h → 0. In this case we prove the existence of spurious solutions Vh of the Galerkin approximation which become unbounded in the maximum norm as h→0. The stability ball Bh then acts to separate the convergent and the spurious solutions. We present the results of some numerical experiments to substantiate our claims.