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The Petrov-Galerkin (PG) method is a robust alternative to the Galerkin method for finite element simulations of challenging partial differential equations (PDEs). The solution of the Galerkin method is obtained from the linear system \(\textbf{B}x=F\) resulting from discretization of trial and test spaces. Although the Galerkin method enforces the equality of trial and test spaces ( \(U_h=V_h\) ) and relies on the inf-sup stability condition to ensure solution accuracy, it often fails for difficult problems where the discrete inf-sup constant \(\alpha _h\) significantly deviates from the abstract inf-sup constant \(\alpha \) . This discrepancy leads to numerical instability and incorrect solutions. The PG method addresses this by allowing distinct trial and test spaces ( \(U_h \ne V_h\) ), enabling the selection of test functions that improve the discrete inf-sup constant \(\alpha _h\) . Of particular interest is the Petrov-Galerkin method with optimal test functions (PGO), where the test functions are computed to maximize \(\alpha _h\) , ensuring stable solutions even for ill-conditioned problems. The PGO method modifies the discrete test space to approximate the abstract stability properties as closely as possible. The computation of optimal test functions involves solving \(\textbf{G} \textbf{W}=\textbf{B}\) , where \(\textbf{B}\) is the Galerkin matrix, and \(\textbf{G}\) is the Gram matrix of the test space’s inner product. Solving \(\textbf{B}^T\textbf{W}x=\textbf{W}^TF\) then yields a stable solution. However, the added computational cost of inverting \(\textbf{G}^{-1}\) poses a significant overhead. In this work, we propose a novel approach that leverages deep neural networks (DNNs) to approximate the inverse of the Gram matrix for a class of advection-diffusion problems with variable diffusion coefficients. By training the DNN to predict \(\textbf{G}^{-1}\) , we eliminate the computational overhead of matrix inversion, enabling efficient and stable solutions of PDEs. Our results demonstrate the effectiveness of the DNN-enhanced PGO method in maintaining stability and accuracy, even for difficult computational problems where the standard Galerkin method fails. This approach represents a significant advancement in the practical applicability of the Petrov-Galerkin framework for solving complex PDEs.

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Augmenting Petrov-Galerkin Method with Optimal Test Functions by DNN Learning the Inverse of the Gram Matrix

  • 会议论文

  • 发表日期：

    2025年

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  • Tomasz Służalec

摘要

The Petrov-Galerkin (PG) method is a robust alternative to the Galerkin method for finite element simulations of challenging partial differential equations (PDEs). The solution of the Galerkin method is obtained from the linear system \(\textbf{B}x=F\) resulting from discretization of trial and test spaces. Although the Galerkin method enforces the equality of trial and test spaces ( \(U_h=V_h\) ) and relies on the inf-sup stability condition to ensure solution accuracy, it often fails for difficult problems where the discrete inf-sup constant \(\alpha _h\) significantly deviates from the abstract inf-sup constant \(\alpha \) . This discrepancy leads to numerical instability and incorrect solutions. The PG method addresses this by allowing distinct trial and test spaces ( \(U_h \ne V_h\) ), enabling the selection of test functions that improve the discrete inf-sup constant \(\alpha _h\) . Of particular interest is the Petrov-Galerkin method with optimal test functions (PGO), where the test functions are computed to maximize \(\alpha _h\) , ensuring stable solutions even for ill-conditioned problems. The PGO method modifies the discrete test space to approximate the abstract stability properties as closely as possible. The computation of optimal test functions involves solving \(\textbf{G} \textbf{W}=\textbf{B}\) , where \(\textbf{B}\) is the Galerkin matrix, and \(\textbf{G}\) is the Gram matrix of the test space’s inner product. Solving \(\textbf{B}^T\textbf{W}x=\textbf{W}^TF\) then yields a stable solution. However, the added computational cost of inverting \(\textbf{G}^{-1}\) poses a significant overhead. In this work, we propose a novel approach that leverages deep neural networks (DNNs) to approximate the inverse of the Gram matrix for a class of advection-diffusion problems with variable diffusion coefficients. By training the DNN to predict \(\textbf{G}^{-1}\) , we eliminate the computational overhead of matrix inversion, enabling efficient and stable solutions of PDEs. Our results demonstrate the effectiveness of the DNN-enhanced PGO method in maintaining stability and accuracy, even for difficult computational problems where the standard Galerkin method fails. This approach represents a significant advancement in the practical applicability of the Petrov-Galerkin framework for solving complex PDEs.