Extension of the primal DPG setting to the parabolic Richards equation

Develop and analyze a primal discontinuous Petrov–Galerkin minimum-residual formulation with least-squares constraints for the parabolic Richards equation, thereby extending the semilinear elliptic framework proposed in the paper to the full time-dependent Richards model.

Background

The semilinear elliptic problems studied here arise from time-stepping discretizations of the Richards equation. The authors argue that, from an analytical perspective, primal DPG formulations are better suited for parabolic problems than ultraweak formulations, suggesting that their primal setting should naturally extend to the parabolic Richards equation.

However, the paper focuses on the elliptic (time-stepped) case, and the full treatment of the parabolic Richards equation within the primal DPG minimum-residual framework is deferred, indicating an open direction for further research.