Numerical analysis of semilinear elliptic equations with finite spectral interaction (Q441950)

The authors explore numerically a global Lyapunov-Schmidt decomposition to obtain a numerical algorithm for solving \(-\Delta u(x)-f(x,u(x))=g(x)\) with Dirichlet boundary conditions in a bounded domain \(\Omega .\) The nonlinearities are non-resonant and have finite spectral interaction: no eigenvalue of \(-\Delta _{D}\) is an endpoint of \(\overline{\partial _{2}f\left( \Omega ,\mathbb{R}\right) }\), which in turn only contains a finite number of eigenvalues (\( \partial _{2}\) is the partial derivative with respect to the second variable). The authors use a class of \(C^{1}\) maps \(F:X\rightarrow Y\) between Banach spaces. Split \(X=W_{X}\oplus V_{X}\) and \(Y=W_{Y}\oplus V_{y}\) into closed horizontal and vertical subspaces. Define complementary projections \(P_{Y},Q_{Y}:Y\rightarrow Y\) such that \(\mathrm {Ran}P_{Y}=W_{Y}\), \(\mathrm {Ran} Q_{Y}=V_{Y}.\) The bifurcation equations related to the decomposition for \( F\left( x\right) =y\) are \(P_{Y}F(w+v)=P_{Y}y,Q_{Y}F(w+v)=Q_{Y}y\), \(w\in W_{X}\), \(v\in V_{X}\). The algorithm first computes \(w\) using the finite element method. The search for solutions of \(F\left( x\right) =y\) then reduces to inverting a computable map between isomorphic finite dimensional subspaces \( V_{X}\) and \(V_{Y}.\)