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Preprint 2020-08

Rommel Bustinza, Jonathan Munguia:

An a priori error analysis for a class of nonlinear elliptic problems applying the hybrid high-order method


In this paper, we analyse a nonlinear elliptic problem in a bounded domain, applying the already known hybrid high-order (HHO) method. Our analysis follows known approaches to deal with diffusion linear problems, and take into account the nonlinear works in elasticity. This approach allows us to obtain high-order approximation of unknowns, by assembling a non-conforming discrete space based on degrees of freedom over the interior volume of each element and on the faces of its boundary, through $L^2$-projections. The principal feature is the use of a gradient reconstruction operator, whose codomain in the current context, is bigger than the considered in the linear case. On the other hand, the stabilization term is the same as for the linear case. In addition, the construction does not depend on the spatial dimension, and it offers the possibility to use general meshes with polytopal cells and non-matching interfaces. The discrete unknowns are chosen such that they are supported over each element and its boundary. It is important to emphasize that the cell-based unknowns can be eliminated locally by a Schur complement technique (also known as static condensation). This allows us to obtain a more compact system, reducing significantly its size, which is solved by Newton's method. We establish the well-posedness of the nonlinear discrete scheme, the stability of the method, and confirm that it is optimally convergent in the energy norm and in the $L^2$-norm, for the potential and its gradient, for smooth enough solutions. In addition, we also establish the convergence of the $L^2$-projection of the potential error and the super convergence of the reconstructive potential error, under the same additional regularity assumption of the exact solution. The latter results, up to the author's knowledge, have not been proven before. Several numerical experiments confirm our theoretical results. We point out that this approach can be also applied/extended to deal with other boundary conditions (such as nonhomogeneous Dirichlet, mixed or pure Neumann).

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