# Numerical Methods for PDEs: In Occasion of Raytcho Lazarov's 70th Birthday

## Leszek Demkowicz, University of Texas at Austin A Tutorial on Discontinuous Petrov Galerkin (DPG) Method with Optimal Test Functions

### Authors:

• Leszek Demkowickz, University of Texas at Austin
• Jay Gopalaksrichan, Portland State University

### Abstract:

We will give a short tutorial on the DPG method emphasizing the main points and illustrating them with numerical examples. Here is a few of them:

1. The DPG method is a minimum-residual method with the residual evaluated in a dual norm.
2. The method can be interpreted as a Petrov-Galerkin method with optimal test functions (realizing the $$\text{sup}$$ in the $$\text{inf}$$-$$\text{sup}$$ condition).
3. The optimal test functions are computed on the fly by inverting approximately) the Riesz operator corresponding to the test space.
4. With broken test spaces and norms capable of localization, the inversion is done element-wise, i.e., the optimal test functions are computed within the element routine. This is more expensive then for the standard finite element method but it is compatible with the standard finite element technology.
5. The main price paid for the localization is the presence of additional unknowns: traces and fluxes. Compared with standard conforming finite element methods or discontinuous Galerkin methods capable of hybridization, the number of (non-local) unknowns doubles and it is of the same range as for discontinuous Galerkin methods. Contrary to discontinuous Galerkin methods based on numerical flux, the flux enters as an additional unknown in the DPG method.
6. The method can be interpreted as a preconditioned least squares method. The stiffness matrix is Hermitian and positive-definite but its condition number is the same as for standard finite elements.
7. The formulation based on a first-order system is very natural but not necessary. You can wish with the second-order equation if you wish. The key point is to break the test functions.
8. There is nothing exotic about the ultra-weak variational formulation behind the DPG method. If the operator is well posed in the $$L^2$$ sense (the operator is $$L^2$$ bounded below), the ultra-weak variational formulation is also well posed with the corresponding $$\text{inf}$$-$$\text{sup}$$ constant being of the same order.
9. With the use of optimal test functions, the issues of approximation and stability are fully separated. This is illustrated by using hp-adaptivity.
10. The method is especially suited for singular perturbation problems, e.g., convection-dominated diffusion, high wave number wave propagation, and elasticity for thin-walled structures. For problems of this type, one can systematically design a test norm to accomplish robustness, i.e., a stability uniform in the perturbation parameter.
11. If you have a hybrid finite element code, converting it to a DPG code is very easy.
12. The methodology extends to nonlinear problems. We will show examples for compressible Navier-Stokes equations.