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We propose an incremental algorithm to compute the proper orthogonal decomposition (POD) of simulation data for a partial differential equation. Specifically, we modify an incremental matrix SVD algorithm of Brand to accommodate data arising from Galerkin-type simulation methods for time dependent PDEs. The algorithm is applicable to data generated by many numerical methods for PDEs, including finite element and discontinuous Galerkin methods. The algorithm initializes and efficiently updates the dominant POD eigenvalues and modes during the time stepping in a PDE solver without storing the simulation data. We prove that the algorithm without truncation updates the POD exactly. We demonstrate the effectiveness of the algorithm using finite element computations for a 1D Burgers’ equation and a 2D Navier–Stokes problem.

We consider a distributed optimal control problem governed by an elliptic convection diffusion PDE, and propose a hybridizable discontinuous Galerkin method to approximate the solution. We use polynomials of degree $$k+1$$ to approximate the state and dual state, and polynomials of degree $$k \ge 0$$ to approximate their fluxes. Moreover, we use polynomials of degree k to approximate the numerical traces of the state and dual state on the faces, which are the only globally coupled unknowns. We prove optimal a priori error estimates for all variables when $$ k \ge 0 $$ . Furthermore, from the point of view of the number of degrees of freedom of the globally coupled unknowns, this method achieves superconvergence for the state, dual state, and control when $$k\ge 1$$ . We illustrate our convergence results with numerical experiments.

We propose the first hybridizable discontinuous Galerkin method for the $p$-Laplacian equation. When using polynomials of degree $k \geq 0$ for the approximation spaces of $u$, $\boldsymbol{\nabla} u$, and $|\boldsymbol{\nabla} u|^{p-2}\boldsymbol{\nabla} u$, the method exhibits optimal $k+1$ order of convergence for all variables in $L^1$- and $L^p$-norms in our numerical experiments. For $k \geq 1$, an elementwise computation allows us to obtain a new approximation $u_h^*$ that converges to $u$ with order $k+2$. We rewrite the scheme as discrete minimization problems in order to solve them with nonlinear minimization algorithms. The unknown of the first problem is the approximation of $u$ on the skeleton of the mesh but requires solving nonlinear local problems. The second problem has the approximation on the elements as an additional unknown but it only requires solving linear local problems. We present numerical results displaying the convergence properties of the methods, demonstrating the utility of...

We propose a new hybridizable discontinuous Galerkin (HDG) model order reduction technique based on proper orthogonal decomposition (POD). We consider the heat equation as a test problem and prove error bounds that converge to zero as the number of POD modes increases. We present 2D and 3D numerical results to illustrate the convergence analysis.

It is now a well known fact that hybridizable discontinuous Galerkin for nonlinear elasticity may not converge to the exact solution if their inter-element jumps are not properly penalized or, equivalently, if the values of their stabilization function are not suitably chosen. For example, if their stabilization function is of order one, the method generates spurious oscillations in the region in which the elastic moduli tensor is indefinite. This phenomenon disappears when the values of the stabilization function are increased. Mixed methods display the same problem, as their stabilization function is identically zero. Here, we find explicit formulas for the lower bound of the values of the stabilization function which allow us to avoid this unpleasant phenomenon. Our numerical experiments show that, when we use polynomials of degree k > 0 for the approximate gradient, first-order Piola-Kirchhoff and displacement, all the approximations converge with the optimal order, k + 1 . They also show that, if we increase the polynomial degree of the local spaces by one and update the values of the stabilization function only for the elements for which the elasticity tensor is indefinite, we obtain that all approximate solutions converge with order k + 1 and that a local post processing of the displacement converges with order k + 2 .

This paper presents a new hybridizable discontinuous Galerkin (HDG) method for linear elasticity on general polyhedral meshes, based on a strong symmetric stress formulation. The key feature of this new HDG method is the use of a special form of the numerical trace of the stresses, which makes the error analysis different from the projection-based error analyzes used for most other HDG methods. For arbitrary polyhedral elements, we approximate the stress by using polynomials of degree k>=1 and the displacement by using polynomials of degree k+1. In contrast, to approximate the numerical trace of the displacement on the faces, we use polynomials of degree k only. This allows for a very efficient implementation of the method, since the numerical trace of the displacement is the only globally-coupled unknown, but does not degrade the convergence properties of the method. Indeed, we prove optimal orders of convergence for both the stresses and displacements on the elements. In the almost incompressible case, we show the error of the stress is also optimal in the standard L2-norm. These optimal results are possible thanks to a special superconvergence property of the numerical traces of the displacement, and thanks to the use of a crucial elementwise Korn's inequality. Several numerical results are presented to support our theoretical findings in the end.