Improved rates for a space–time FOSLS of parabolic PDEs

Abstract

Abstract We consider the first-order system space–time formulation of the heat equation introduced by Bochev and Gunzburger (in: Bochev and Gunzburger (eds) Applied mathematical sciences, vol 166, Springer, New York, 2009), and analyzed by Führer and Karkulik (Comput Math Appl 92:27–36, 2021) and Gantner and Stevenson (ESAIM Math Model Numer Anal 55(1):283–299 2021), with solution components $$(u_1,\textbf{u}_2)=(u,-\nabla _\textbf{x} u)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mrow> <mml:mo>(</mml:mo> <mml:msub> <mml:mi>u</mml:mi> <mml:mn>1</mml:mn> </mml:msub> <mml:mo>,</mml:mo> <mml:msub> <mml:mi>u</mml:mi> <mml:mn>2</mml:mn> </mml:msub> <mml:mo>)</mml:mo> </mml:mrow> <mml:mo>=</mml:mo> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>u</mml:mi> <mml:mo>,</mml:mo> <mml:mo>-</mml:mo> <mml:msub> <mml:mi>∇</mml:mi> <mml:mi>x</mml:mi> </mml:msub> <mml:mi>u</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:mrow> </mml:math> . The corresponding operator is boundedly invertible between a Hilbert space U and a Cartesian product of $$L_2$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>L</mml:mi> <mml:mn>2</mml:mn> </mml:msub> </mml:math> -type spaces, which facilitates easy first-order system least-squares (FOSLS) discretizations. Besides $$L_2$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>L</mml:mi> <mml:mn>2</mml:mn> </mml:msub> </mml:math> -norms of $$\nabla _\textbf{x} u_1$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mi>∇</mml:mi> <mml:mi>x</mml:mi> </mml:msub> <mml:msub> <mml:mi>u</mml:mi> <mml:mn>1</mml:mn> </mml:msub> </mml:mrow> </mml:math> and $$\textbf{u}_2$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>u</mml:mi> <mml:mn>2</mml:mn> </mml:msub> </mml:math> , the (graph) norm of U contains the $$L_2$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>L</mml:mi> <mml:mn>2</mml:mn> </mml:msub> </mml:math> -norm of $$\partial _t u_1 +{{\,\textrm{div}\,}}_\textbf{x} \textbf{u}_2$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mi>∂</mml:mi> <mml:mi>t</mml:mi> </mml:msub> <mml:msub> <mml:mi>u</mml:mi> <mml:mn>1</mml:mn> </mml:msub> <mml:mo>+</mml:mo> <mml:msub> <mml:mrow> <mml:mspace/> <mml:mtext>div</mml:mtext> <mml:mspace/> </mml:mrow> <mml:mi>x</mml:mi> </mml:msub> <mml:msub> <mml:mi>u</mml:mi> <mml:mn>2</mml:mn> </mml:msub> </mml:mrow> </mml:math> . When applying standard finite elements w.r.t. simplicial partitions of the space–time cylinder, estimates of the approximation error w.r.t. the latter norm require higher-order smoothness of $$\textbf{u}_2$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>u</mml:mi> <mml:mn>2</mml:mn> </mml:msub> </mml:math> . In experiments for both uniform and adaptively refined partitions, this manifested itself in disappointingly low convergence rates for non-smooth solutions u . In this paper, we construct finite element spaces w.r.t. prismatic partitions. They come with a quasi-interpolant that satisfies a near commuting diagram in the sense that, apart from some harmless term, the aforementioned error depends exclusively on the smoothness of $$\partial _t u_1 +{{\,\textrm{div}\,}}_\textbf{x} \textbf{u}_2$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mi>∂</mml:mi> <mml:mi>t</mml:mi> </mml:msub> <mml:msub> <mml:mi>u</mml:mi> <mml:mn>1</mml:mn> </mml:msub> <mml:mo>+</mml:mo> <mml:msub> <mml:mrow> <mml:mspace/> <mml:mtext>div</mml:mtext> <mml:mspace/> </mml:mrow> <mml:mi>x</mml:mi> </mml:msub> <mml:msub> <mml:mi>u</mml:mi> <mml:mn>2</mml:mn> </mml:msub> </mml:mrow> </mml:math> , i.e., of the forcing term $$f=(\partial _t-\Delta _x)u$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>f</mml:mi> <mml:mo>=</mml:mo> <mml:mo>(</mml:mo> <mml:msub> <mml:mi>∂</mml:mi> <mml:mi>t</mml:mi> </mml:msub> <mml:mo>-</mml:mo> <mml:msub> <mml:mi>Δ</mml:mi> <mml:mi>x</mml:mi> </mml:msub> <mml:mo>)</mml:mo> <mml:mi>u</mml:mi> </mml:mrow> </mml:math> . Numerical results show significantly improved convergence rates.

Topics & Concepts