A Mass-Preserving Two-Step Lagrange–Galerkin Scheme for Convection-Diffusion Problems

Journal of Scientific Computing, vol. 92

Abstract

Abstract A mass-preserving two-step Lagrange–Galerkin scheme of second order in time for convection-diffusion problems is presented, and convergence with optimal error estimates is proved in the framework of $$L^2$$ L 2 -theory. The introduced scheme maintains the advantages of the Lagrange–Galerkin method, i.e., CFL-free robustness for convection-dominated problems and a symmetric and positive coefficient matrix resulting from the discretization. In addition, the scheme conserves the mass on the discrete level if the involved integrals are computed exactly. Unconditional stability and error estimates of second order in time are proved by employing two new key lemmas on the truncation error of the material derivative in conservative form and on a discrete Gronwall inequality for multistep methods. The mass-preserving property is achieved by the Jacobian multiplication technique introduced by Rui and Tabata in 2010, and the accuracy of second order in time is obtained based on the idea of the multistep Galerkin method along characteristics originally introduced by Ewing and Russel in 1981. For the first time step, the mass-preserving scheme of first order in time by Rui and Tabata in 2010 is employed, which is efficient and does not cause any loss of convergence order in the $$\ell ^\infty (L^2)$$ ℓ ∞ ( L 2 ) - and $$\ell ^2(H^1_0)$$ ℓ 2 ( H 0 1 ) -norms. For the time increment $$\Delta t$$ Δ t , the mesh size h and a conforming finite element space of polynomial degree $$k \in {\mathbb {N}}$$ k ∈ N , the convergence order is of $$O(\Delta t^2 + h^k)$$ O ( Δ t 2 + h k ) in the $$\ell ^\infty (L^2)\cap \ell ^2(H^1_0)$$ ℓ ∞ ( L 2 ) ∩ ℓ 2 ( H 0 1 ) -norm and of $$O(\Delta t^2 + h^{k+1})$$ O ( Δ t 2 + h k + 1 ) in the $$\ell ^\infty (L^2)$$ ℓ ∞ ( L 2 ) -norm if the duality argument can be employed. Error estimates of $$O(\Delta t^{3/2}+h^k)$$ O ( Δ t 3 / 2 + h k ) in discrete versions of the $$L^\infty (H^1_0)$$ L ∞ ( H 0 1 ) - and $$H^1(L^2)$$ H 1 (

Authors 4

1. Kouta Futai

   Panasonic (Japan) · Panasonic System Networks Co., Ltd. (Japan)

   Affiliation as printed

   Panasonic System Networks R &D Lab. Co., Ltd., Sainen 1-1-3, Kanazawa, 920-0024, Japan

2. Niklas Kolbe corresponding Aachen Institute of Geometry and Practical Mathematics

   RWTH Aachen University

   Affiliation as printed

   Institute of Geometry and Practical Mathematics, RWTH Aachen University, Templergraben 55, 52062, Aachen, Germany

   Institute of Geometry and Practical Mathematics, RWTH Aachen University, Templergraben 55, 52062, Aachen, Germany

3. Hirofumi Notsu

   Kanazawa University

   Affiliation as printed

   Faculty of Mathematics and Physics, Kanazawa University, Kakuma, Kanazawa, 920-1192, Japan

4. Tasuku Suzuki

   Kanazawa University

   Affiliation as printed

   Division of Mathematical and Physical Sciences, Kanazawa University, Kakuma, Kanazawa, 920-1192, Japan