Unstabilized Hybrid High-Order method for a class of degenerate convex minimization problems
A talk in the BI.discrete series by
Tien Tran Ngoc from Humboldt-Universität Berlin
| Abstract: | The relaxation in the calculus of variation motivates the numerical analysis of a class of degenerate convex minimization problems
with non-strictly convex energy densities with some convexity
control and two-sided $p$-growth. The minimizers may be non-unique in the primal variable but lead to a unique stress $\sigma \in
H(\operatorname{div},\Omega;\mathbb{M})$. Examples include the p-Laplacian,
an optimal design problem in topology optimization, and the convexified
double-well problem.
The approximation by hybrid high-order methods (HHO) utilizes a
reconstruction of the gradients with piecewise Raviart-Thomas finite
elements without stabilization on a regular triangulation into simplices.
The application of this HHO method to the class of degenerate convex
minimization problems allows for a unique $H(\operatorname{div})$ conforming
stress approximation $\sigma_h$. The main results are a-priori
and a posteriori error estimates for the stress error $\sigma-\sigma_h$ in
Lebesgue norms and a computable lower energy bound. Numerical benchmarks
display higher convergence rates for higher polynomial degrees and include
adaptive mesh-refining with the first superlinear convergence rates of
guaranteed lower energy bounds. Zoom Meeting ID: 92653100938 Passcode: 1928 |