Hybridizable Staggered Discontinuous Galerkin Methods for Polyharmonic Equations on Polytopes
This work provides a unified numerical framework for high-order polyharmonic equations on general polytopal meshes, addressing a known bottleneck in mesh flexibility and stability.
The paper develops hybridizable staggered discontinuous Galerkin methods for arbitrary-order polyharmonic equations on polytopal meshes, proving well-posedness and optimal energy error estimates, with numerical experiments confirming predicted convergence rates.
Hybridizable staggered discontinuous Galerkin methods are developed for arbitrary-order polyharmonic equations $(-Δ)^m u=f$ on shape-regular polytopal meshes in $\mathbb R^d$, for any $m\ge1$, $d\ge2$, and polynomial degree $k\ge0$. The method uses the mixed variable $σ=\nabla^m u$ and a staggered primal--dual mesh to impose complementary continuity on scalar and tensor unknowns, without restrictions such as $d\ge m$. Local trace and bubble enrichments stabilize low-order tensor spaces without adding global unknowns. Hybridization localizes the tensor variable and yields an equivalent stabilization-free weak Galerkin formulation. Well-posedness and optimal energy error estimates are proved, and numerical experiments on polygonal and tetrahedral meshes confirm the predicted rates.