| Abstract: |
| We propose and analyse $H^2$-conforming arbitrary order Virtual Element Methods (VEM) for the numerical solution of uniformly elliptic Isaacs equations, a general class of fully nonlinear equations which includes the Hamilton-Jacobi-Bellman as a special case. The approach builds upon our previous work on $H^2$-conforming VEM for linear elliptic problems in nondivergence form, extending the methodology to the fully nonlinear setting. A key feature of the proposed method is that the use of $H^2$-conforming spaces enables a direct, arbitrary order, discretisation of second-order operators in strong form. Moreover, the direct discretization of the strong form allows for a simple weak imposition of the boundary conditions. Assuming that the differential operator satisfies the Cordes conditions, we establish the wellposedness of the VEM and derive optimal a priori error bounds in the $H^2$-norm. Numerical experiments confirm the optimality of the method and its competitiveness with other approaches. |
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