Search for: All records

We propose and analyze a two-scale finite element method for the Isaacs equation. The fine scale is given by the mesh size h whereas the coarse scale ε is dictated by an integro-differential approximation of the partial differential equation. We show that the method satisfies the discrete maximum principle provided that the mesh is weakly acute. This, in conjunction with weak operator consistency of the finite element method, allows us to establish convergence of the numerical solution to the viscosity solution as ε , h → 0, and ε  ≳ ( h |log h |) 1/2 . In addition, using a discrete Alexandrov Bakelman Pucci estimate we deduce rates of convergence, under suitable smoothness assumptions on the exact solution. 

We develop discrete $$W^2_p$$-norm error estimates for the Oliker--Prussner method applied to the Monge--Ampère equation. This is obtained by extending discrete Alexandroff estimates and showing that the contact set of a nodal function contains information on its second-order difference. In addition, we show that the size of the complement of the contact set is controlled by the consistency of the method. Combining both observations, we show that the error estimate $$\| u - u_h \|_{W^2_{f,p}} (\mathcal{N}^I_h)$$ converges in order $$O(h^{1/p})$$ if $p > d$ and converges in order $$O(h^{1/d} \ln (\frac 1 h)^{1/d})$$ if $$p \leq d$$, where $$\|\cdot\|_{W^2_{f,p}(\mathcal{N}^I_h)}$$ is a weighted $$W^2_p$$-type norm, and the constant $C>0$ depends on $$\|{u}\|_{C^{3,1}(\bar\Omega)}$$, the dimension $$d$$, and the constant $$p$$. Numerical examples are given in two space dimensions and confirm that the estimate is sharp in several cases.