A hybrid high-order method for the biharmonic problem
This work addresses numerical analysis challenges in computational mathematics, particularly for solving biharmonic problems with singular solutions, though it appears incremental as it builds on existing hybrid high-order methodology.
The paper tackles the biharmonic and eigenvalue problems by proposing a new hybrid high-order discretization that includes additional degrees of freedom in lower-dimensional submanifolds, enabling a commuting property in any space dimension and resulting in guaranteed lower eigenvalue bounds of higher order, quasi-best approximation estimates, and reliable a posteriori error estimators.
This paper proposes a new hybrid high-order discretization for the biharmonic problem and the corresponding eigenvalue problem. The discrete ansatz space includes degrees of freedom in $n-2$ dimensional submanifolds (e.g., nodal values in 2D and edge values in 3D), in addition to the typical degrees of freedom in the mesh and on the hyperfaces in the HHO literature. This approach enables the characteristic commuting property of the hybrid high-order methodology in any space dimension. The main results are guaranteed lower eigenvalue bounds of higher order. Furthermore, we derive quasi-best approximation estimates as well as reliable and efficient a~posteriori error estimators under minimal regularity assumptions on the exact solution. The latter motivates an adaptive mesh-refining algorithm that empirically recovers optimal convergence rates for singular solutions.