Maciej Paszynski

Department of Computer Science, AGH University of Science and Technology, Krakow, Poland

The finite element method (FEM) is a popular tool for solving engineering problems governed by Partial Differential Equations (PDEs).

The accuracy of the numerical solution depends on the quality of the computational mesh. We consider the self-adaptive hp-FEM, which generates optimal mesh refinements and delivers exponential convergence of the numerical error with respect to the mesh size [1,2]. Thus, it enables solving difficult engineering problems with the highest possible numerical accuracy. The original self-adaptive hp-FEM algorithm proposed by [1] iteratively selects the refinements. It first selects the optimal refinements for element edges. The algorithm of determining the optimal edge refinements consists of two steps. Step 1 is local: It determines the optimal refinement for each edge. Step 2 is global: Given the optimal refinements for edges, it determines which edges to refine and how many unknowns to invest. Once the optimal edge refinements have been determined, they set up a stage for determining optimal refinements for elements. For instance, all elements adjacent to an h-refined edge haveto be h-refined. The polynomial orders on the edges are already known.

Determining optimal element refinements includes now again two steps: local and global. In [3] we propose the reversed approach. We introduce the simplified, one step, algorithm, as a kernel for the selection of the optimal refinements for the interiors of elements. The edge refinements follows the refinements patterns of the interiors, and the polynomial orders on edge are adjusted by taking the minimum of the corresponding orders of interiors. The reversed algorithm allows to train the Deep Neural Network (DNN) to make the optimal decisions on the refinements of element interiors. Thus, we replace the computationally complex kernel of the refinement algorithm by a DNN. The network learns how to optimally refine the elements and modify the orders of the polynomials. In this way, the deterministic algorithm is replaced by a neural network that selects similar quality refinements. We verify our algorithm on a model L-shape domain problem.