A painless multi-level automatic goal-oriented hp-adaptive coarsening strategy for elliptic and non-elliptic problems

This work extends an automatic energy-norm $hp$-adaptive strategy based on performing quasi-optimal unrefinements to the case of non-elliptic problems and goal-oriented adaptivity. The proposed approach employs a multi-level hierarchical data structure and alternates global $h$- and $p$-refinements with a coarsening step. Thus, at each unrefinement step, we eliminate the basis functions with the lowest contributions to the solution. When solving elliptic problems using energy-norm adaptivity, the removed basis functions are those with the lowest contributions to the energy of the solution. For non-elliptic problems or goal-oriented adaptivity, we propose an upper bound of the error representation expressed in terms of an inner product of the specific equation, leading to error indicators that deliver quasi-optimal $hp$-unrefinements. This unrefinement strategy removes unneeded unknowns possibly introduced during the pre-asymptotical regime. In addition, the grids over which we perform the unrefinements are arbitrary, and thus, we can limit their size and associated computational costs. We numerically analyze our algorithm for energy-norm and goal-oriented adaptivity. In particular, we solve two-dimensional ($2$D) Poisson, Helmholtz, convection-dominated equations, and a three-dimensional ($3$D) Helmholtz-like problem. In all cases, we observe \revb{exponential} convergence rates. Our algorithm is robust and straightforward to implement; therefore, it can be easily adapted for industrial applications.BERC.2022-202

Easy-to-implement hp-adaptivity for non-elliptic goal-oriented problems

The FEM has become a foundational numerical technique in computational mechanics and civil engineering since its inception by Courant in 1943 Courant1943. Originating from the Ritz method and variational calculus, the FEM was primarily employed to derive solutions for vibrational systems. A distinctive strength of the FEM is its capability to represent mathematical models through the weak variational formulation of PDE, facilitating computational feasibility even in intricate geometries. However, the search for accuracy often imposes a significant computational task. In the FEM, adaptive methods have emerged to balance the accuracy of solutions with computational costs. The $h$-adaptive FEM designs more efficient meshes by reducing the mesh size $h$ locally while keeping the polynomial order of approximation $p$ fixed (usually $p=1,2$). An alternative approach to the $h$-adaptive FEM is the $p$-adaptive FEM, which locally enriches the polynomial space $p$ while keeping the mesh size $h$ constant. By dynamically adapting $h$ and $p$, the $hp$-adaptive FEM achieves exponential convergence rates. Adaptivity is crucial for obtaining accurate solutions. However, the traditional focus on global norms, such as $L^2$ or $H^1$, might only sometimes serve the requirements of specific applications. In engineering, controlling errors in specific domains related to a particular QoI is often more critical than focusing on the overall solution. That motivated the development of GOA strategies. In this dissertation, we develop automatic GO $hp$-adaptive algorithms tailored for non-elliptic problems. These algorithms shine in terms of robustness and simplicity in their implementation, attributes that make them especially suitable for industrial applications. A key advantage of our methodologies is that they do not require computing reference solutions on globally refined grids. Nevertheless, our approach is limited to anisotropic $p$ and isotropic $h$ refinements. We conduct multiple tests to validate our algorithms. We probe the convergence behavior of our GO $h$- and $p$-adaptive algorithms using Helmholtz and convection-diffusion equations in one-dimensional scenarios. We test our GO $hp$-adaptive algorithms on Poisson, Helmholtz, and convection-diffusion equations in two dimensions. We use a Helmholtz-like scenario for three-dimensional cases to highlight the adaptability of our GO algorithms. We also create efficient ways to build large databases ideal for training DNN using $hp$ MAGO FEM. As a result, we efficiently generate large databases, possibly containing hundreds of thousands of synthetic datasets or measurements