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An Interpolated Galerkin Finite Element Method for the Poisson Equation

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Abstract

We develop a new approach to construct finite element methods to solve the Poisson equation. The idea is to use the pointwise Laplacian as a degree of freedom followed by interpolating the solution at the degree of freedom by the given right-hand side function in the partial differential equation. The finite element solution is then the Galerkin projection in a smaller vector space. This idea is similar to that of interpolating the boundary condition in the standard finite element method. Our approach results in a smaller system of equations and of a better condition number. The number of unknowns on each element is reduced significantly from \((k^2+3k+2)/2\) to 3k for the \(P_k\) (\(k\ge 3\)) finite element. We construct bivariate \(P_2\) conforming and nonconforming, and \(P_k\) (\(k\ge 3\)) conforming interpolated Galerkin finite elements on triangular grids; prove their optimal order of convergence; and confirm our findings by numerical tests.

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Authors and Affiliations

  1. Department of Mathematics, Towson University, 7800 York Road, Towson, MD, 21252, USA

    Tatyana Sorokina

  2. Department of Mathematical Sciences, University of Delaware, Newark, DE, 19716, USA

    Shangyou Zhang

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  1. Tatyana Sorokina
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  2. Shangyou Zhang
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Correspondence to Tatyana Sorokina.

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Sorokina, T., Zhang, S. An Interpolated Galerkin Finite Element Method for the Poisson Equation. J Sci Comput 92, 47 (2022). https://doi.org/10.1007/s10915-022-01903-x

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  • DOI: https://doi.org/10.1007/s10915-022-01903-x

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