Adaptive Least-Squares (Space-Time) Finite Element Methods For Convection-Diffusion Problems

Abstract In this paper we formulate and analyze adaptive (space-time) least-squares finite element methods for the solution of convection-diffusion equations. The convective derivative <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>𝒗</m:mi> <m:mo>⋅</m:mo> <m:mrow> <m:mo>∇</m:mo> <m:mo>⁡</m:mo> <m:mi>u</m:mi> </m:mrow> </m:mrow> </m:math> {\boldsymbol{v}\cdot\nabla u} is considered as part of the total time derivative <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mrow> <m:mfrac> <m:mi>d</m:mi> <m:mrow> <m:mi>d</m:mi> <m:mo>⁢</m:mo> <m:mi>t</m:mi> </m:mrow> </m:mfrac> <m:mo>⁢</m:mo> <m:mi>u</m:mi> </m:mrow> <m:mo>=</m:mo> <m:mrow> <m:mrow> <m:msub> <m:mo>∂</m:mo> <m:mi>t</m:mi> </m:msub> <m:mo>⁡</m:mo> <m:mi>u</m:mi> </m:mrow> <m:mo>+</m:mo> <m:mrow> <m:mi>𝒗</m:mi> <m:mo>⋅</m:mo> <m:mrow> <m:mo>∇</m:mo> <m:mo>⁡</m:mo> <m:mi>u</m:mi> </m:mrow> </m:mrow> </m:mrow> </m:mrow> </m:math> {\frac{d}{dt}u=\partial_{t}u+\boldsymbol{v}\cdot\nabla u} , and therefore we can use a rather standard stability and error analysis for related space-time finite element methods. For stationary problems we restrict the ansatz space <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:msubsup> <m:mi>H</m:mi> <m:mn>0</m:mn> <m:mn>1</m:mn> </m:msubsup> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi mathvariant="normal">Ω</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:math> {H^{1}_{0}(\Omega)} such that the convective derivative is considered as an element of the dual <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:msup> <m:mi>H</m:mi> <m:mrow> <m:mo>-</m:mo> <m:mn>1</m:mn> </m:mrow> </m:msup> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi mathvariant="normal">Ω</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:math> {H^{-1}(\Omega)} of the test space <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:msubsup> <m:mi>H</m:mi> <m:mn>0</m:mn> <m:mn>1</m:mn> </m:msubsup> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi mathvariant="normal">Ω</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:math> {H^{1}_{0}(\Omega)} , which also allows unbounded velocities <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>𝒗</m:mi> </m:math> {\boldsymbol{v}} . While the discrete finite element schemes are always unique solvable, the numerical solutions may suffer from a bad approximation property of the finite element space when considering convection dominated problems, i.e., small diffusion coefficients. Instead of adding suitable stabilization terms, we aim to resolve the solutions by using adaptive (space-time) finite element methods. For this we introduce a least-squares approach where the discrete adjoint defines local a posteriori error indicators to drive an adaptive scheme. Numerical examples illustrate the theoretical considerations.