A Superconvergent Ensemble HDG Method for Parameterized Convection Diffusion Equations

Abstract

We devised a first order time stepping ensemble hybridizable discontinuous Galerkin (HDG) method for a group of parameterized convection diffusion PDEs with different initial conditions, body forces, boundary conditions and coefficients in our earlier work [3]. We obtained an optimal convergence rate for the ensemble solutions in L ∞ (0, T ; L 2 (Ω)) on a simplex mesh; and obtained a superconvergent rate for the ensemble solutions in L 2 (0, T ; L 2 (Ω)) after an elementby-element postprocessing if polynomials degree k ≥ 1 and the coefficients of the PDEs are independent of time. In this work, we propose a new second order time stepping ensemble HDG method to revisit the problem. We obtain a superconvergent rate for the ensemble solutions in L ∞ (0, T ; L 2 (Ω)) without an element-by-element postprocessing for all polynomials degree k ≥ 0. Furthermore, our mesh can be any polyhedron, no need to be simplex; and the coefficients of the PDEs can dependent on time. Finally, we present numerical experiments to confirm our theoretical results.

A New Ensemble HDG Method for Parameterized Convection Diffusion PDEs

“…(4.1) with respect to time t. It is worth mentioning that we do not need to assume that the coefficients are independent of time. However, we need to assume the coefficients are independent of time in the previous work [3]. To shorten lengthy equations, we define the following HDG operators B j and C j :…”

“…To the best of our knowledge, all previous works only contain a suboptimal L ∞ (0, T ; L 2 (Ω)) convergent rate for the ensemble solutions u j . Only one other very recent work [3] contains an optimal L ∞ (0, T ; L 2 (Ω)) convergent rate for the ensemble solutions u j , and a L 2 (0, T ; L 2 (Ω)) superconvergent rate if the coefficients of the PDEs are independent of time and degree polynomial k ≥ 1; our main result: Theorem 4.2 is the first time to obtain the L ∞ (0, T ; L 2 (Ω)) supconvergent rate for the ensemble solutions u j for all k ≥ 0 and without assume that the coefficients of the PDEs are independent of time. It is also the first time to obtain the superconvergent rate for a single convection diffusion PDE when k = 0.…”

“…More recently, we proposed a first order time stepping ensemble hybridizable discontinuous Galerkin (HDG) method in [3] to study a group of convection diffusion PDEs with different initial conditions, boundary conditions, body forces and coefficients. We obtained an optimal L ∞ (0, T ; L 2 (Ω)) convergence rate for the solutions on a simplex mesh, and we obtained a L 2 (0, T ; L 2 (Ω)) superconvergent rate if the polynomials of degree k ≥ 1 and the coefficients of the PDEs are independent of time.…”

A new reduced order model of linear parabolic PDEs

“…This can be computationally expensive if m is not small. Following an ensemble idea in [6], we can treat {b i } m i=0 simultaneously (see Algorithm 4) since these linear systems share a common coefficient matrix; it is more efficient than solving the linear system with a single RHS for m + 1 times.…”

Convergence Analysis and Cost Estimate of an MLMC-HDG Method for Elliptic PDEs with Random Coefficients

“…The error analysis of elliptic PDEs was given by introducing a projection (see, e.g., [17] and references therein). To date, the HDG method has been successfully applied to parabolic, convection diffusion problems, phase flows and optimal control problem (see, e.g., [18][19][20][21][22]). However, we did not find theoretical or numerical analysis works on the HDG method for PDEs with random coefficients.…”