ℎ𝑝-Discontinuous Galerkin methods for the Helmholtz equation with large wave number

Abstract

In this paper we develop and analyze some interior penalty <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="h p"> <mml:semantics> <mml:mrow> <mml:mi>h</mml:mi> <mml:mi>p</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">hp</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -discontinuous Galerkin ( <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="h p"> <mml:semantics> <mml:mrow> <mml:mi>h</mml:mi> <mml:mi>p</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">hp</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -DG) methods for the Helmholtz equation with first order absorbing boundary condition in two and three dimensions. The proposed <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="h p"> <mml:semantics> <mml:mrow> <mml:mi>h</mml:mi> <mml:mi>p</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">hp</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -DG methods are defined using a sesquilinear form which is not only mesh-dependent (or <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="h"> <mml:semantics> <mml:mi>h</mml:mi> <mml:annotation encoding="application/x-tex">h</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -dependent) but also degree-dependent (or <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="p"> <mml:semantics> <mml:mi>p</mml:mi> <mml:annotation encoding="application/x-tex">p</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -dependent). In addition, the sesquilinear form contains penalty terms which not only penalize the jumps of the function values across the element edges but also the jumps of the first order tangential derivatives as well as jumps of all normal derivatives up to order <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="p"> <mml:semantics> <mml:mi>p</mml:mi> <mml:annotation encoding="application/x-tex">p</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . Furthermore, to ensure the stability, the penalty parameters are taken as complex numbers with positive imaginary parts, so essentially and practically no constraint is imposed on the penalty parameters. It is proved that the proposed <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="h p"> <mml:semantics> <mml:mrow> <mml:mi>h</mml:mi> <mml:mi>p</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">hp</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -discontinuous Galerkin methods are stable (hence, well-posed) without any mesh constraint. For each fixed wave number <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> , sub-optimal order (with respect to <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="h"> <mml:semantics> <mml:mi>h</mml:mi> <mml:annotation encoding="application/x-tex">h</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="p"> <mml:semantics> <mml:mi>p</mml:mi> <mml:annotation encoding="application/x-tex">p</mml:annotation> </mml:semantics> </mml:math> </inline-formula> ) error estimates in the broken <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper H Superscript 1"> <mml:semantics> <mml:msup> <mml:mi>H</mml:mi> <mml:mn>1</mml:mn> </mml:msup> <mml:annotation encoding="application/x-tex">H^1</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -norm and the <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">L^2</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -norm are derived without any mesh constraint. The error estimates as well as the stability estimates are improved to optimal order under the mesh condition <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="k cubed h squared p Superscript negative 2 Baseline less-than-or-equal-to upper C 0"> <mml:semantics> <mml:mrow> <mml:msup> <mml:mi>k</mml:mi> <mml:mn>3</mml:mn> </mml:msup> <mml:msup> <mml:mi>h</mml:mi> <mml:mn>2</mml:mn> </mml:msup> <mml:msup> <mml:mi>p</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo> − </mml:mo> <mml:mn>2</mml:mn> </mml:mrow> </mml:msup> <mml:mo> ≤ </mml:mo> <mml:msub> <mml:mi>C</mml:mi> <mml:mn>0</mml:mn> </mml:msub> </mml:mrow> <mml:annotation encoding="application/x-tex">k^3h^2p^{-2}\le C_0</mml:annotation> </mml:semantics> </mml:math> </inline-formula> by utilizing these stability and error estimates and using a stability-error iterative procedure, where <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper C 0"> <mml:semantics> <mml:msub> <mml:mi>C</mml:mi> <mml:mn>0</mml:mn> </mml:msub> <mml:annotation encoding="application/x-tex">C_0</mml:annotation> </mml:semantics> </mml:math> </inline-formula>