2015/02/17 by Hailiang Liu · 68 citations
Engineering · Mathematics · #Advanced Numerical Methods in Computational Mathematics #Algorithm #Applied mathematics #Cartesian coordinate system #Computational Fluid Dynamics and Aerodynamics #Convection #Convection–diffusion equation #Degree (music) #Degree of a polynomial #Differential Equations and Numerical Methods #Discontinuous Galerkin method #Finite element method #Galerkin method #Geometry #Jump #Mathematical analysis #Mathematics #Polynomial #Projection (relational algebra) #Superconvergence
paper · pdf · doi:10.1090/s0025-5718-2015-02923-8
published in Mathematics of Computation 84(295), 2263-2295 (American Mathematical Society)
openalex publication_date 2015/02/17 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/06
In this paper, we present the optimal <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper L squared"> <mml:semantics> <mml:msup> <mml:mi>L</mml:mi> <mml:mn>2</mml:mn> </mml:msup> <mml:annotation encoding="application/x-tex">L2</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -error estimate of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper O left-parenthesis h Superscript k plus 1 Baseline right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi>O</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:msup> <mml:mi>h</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>k</mml:mi> <mml:mo>+</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:msup> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">O(hk+1)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> for polynomial elements of degree <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="k"> <mml:semantics> <mml:mi>k</mml:mi> <mml:annotation encoding="application/x-tex">k</mml:annotation> </mml:semantics> </mml:math> </inline-formula> of the semidiscrete direct discontinuous Galerkin method for convection-diffusion equations. The main technical difficulty lies in the control of the inter-element jump terms which arise because of the convection and the discontinuous nature of numerical solutions. The main idea is to use some global projections satisfying interface conditions dictated by the choice of numerical fluxes so that trouble terms at the cell interfaces are eliminated or controlled. In multi-dimensional case, the orders of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="k plus 1"> <mml:semantics> <mml:mrow> <mml:mi>k</mml:mi> <mml:mo>+</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">k+1</mml:annotation> </mml:semantics> </mml:math> </inline-formula> hinge on a superconvergence estimate when tensor product polynomials of degree <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="k"> <mml:semantics> <mml:mi>k</mml:mi> <mml:annotation encoding="application/x-tex">k</mml:annotation> </mml:semantics> </mml:math> </inline-formula> are used on Cartesian grids. A collection of projection errors in both one- and multi-dimensional cases is established.