Simultaneous approximation terms and functional accuracy for diffusion problems discretized with multidimensional summation-by-parts operators

by   Zelalem Arega Worku, et al.

Several types of simultaneous approximation term (SAT) for diffusion problems discretized with multidimensional summation-by-parts (SBP) operators are analyzed based on a common framework. Conditions under which the SBP-SAT discretizations are consistent, conservative, adjoint consistent, and energy stable are presented. For SATs leading to primal and adjoint consistent discretizations, the error in output functionals is shown to be of order h^2p when a degree p multidimensional SBP operator is used to discretize the spatial derivatives. SAT penalty coefficients corresponding to various discontinuous Galerkin fluxes developed for elliptic partial differential equations are identified. We demonstrate that the original method of Bassi and Rebay, the modified method of Bassi and Rebay, and the symmetric interior penalty method are equivalent when implemented with SBP diagonal-E operators that have diagonal norm matrix, e.g., the Legendre-Gauss-Lobatto SBP operator in one space dimension. Similarly, the local discontinuous Galerkin and the compact discontinuous Galerkin schemes are equivalent for this family of operators. The analysis remains valid on curvilinear grids if a degree ≤ p+1 bijective polynomial mapping from the reference to physical elements is used. Numerical experiments with the two-dimensional Poisson problem support the theoretical results.



There are no comments yet.


page 1

page 2

page 3

page 4


Stability and Functional Superconvergence of Narrow-Stencil Second-Derivative Generalized Summation-By-Parts Discretizations

We analyze the stability and functional superconvergence of discretizati...

A mixed discontinuous Galerkin method without interior penalty for time-dependent fourth order problems

A novel discontinuous Galerkin (DG) method is developed to solve time-de...

Should multilevel methods for discontinuous Galerkin discretizations use discontinuous interpolation operators?

Multi-level preconditioners for Discontinuous Galerkin (DG) discretizati...

Superconvergence of Discontinuous Galerkin methods for Elliptic Boundary Value Problems

In this paper, we present a unified analysis of the superconvergence pro...

A hybridized discontinuous Galerkin method for Poisson-type problems with sign-changing coefficients

In this paper, we present a hybridized discontinuous Galerkin (HDG) meth...

Families of hybridizable interior penalty discontinuous Galerkin methods for degenerate advection-diffusion-reaction problems

We analyze families of primal high-order hybridizable discontinuous Gale...

Robust interior penalty discontinuous Galerkin methods

Classical interior penalty discontinuous Galerkin (IPDG) methods for dif...
This week in AI

Get the week's most popular data science and artificial intelligence research sent straight to your inbox every Saturday.