The Convection-Diffusion-Reaction Equation in Non-Hilbert Sobolev Spaces: A Direct Proof of the Inf-Sup Condition and Stability of Galerkin’s Method

While it is classical to consider the solution of the convection-diffusion-reaction equation in the Hilbert space H 0 1 ⁢ ( Ω ) {H_{0}^{1}(\Omega)} , the Banach Sobolev space W 0 1 , q ⁢ ( Ω ) {W^{1,q}_{0}(\Omega)} , 1 < q < ∞ {1 , is more general allowing more irregular solutions. In this paper we present a well-posedness theory for the convection-diffusion-reaction equation in the W 0 1 , q ⁢ ( Ω ) {W^{1,q}_{0}(\Omega)} - W 0 1 , q ′ ⁢ ( Ω ) {W_{0}^{1,q^{\prime}}(\Omega)} functional setting, 1 q + 1 q ′ = 1 {\frac{1}{q}+\frac{1}{q^{\prime}}=1} . The theory is based on directly establishing the inf-sup conditions. Apart from a standard assumption on the advection and reaction coefficients, the other key assumption pertains to a subtle regularity requirement for the standard Laplacian. An elementary consequence of the well-posedness theory is the stability and convergence of Galerkin’s method in this setting, for a diffusion-dominated case and under the assumption of W 1 , q ′ {W^{1,q^{\prime}}} -stability of the H 0 1 {H_{0}^{1}} -projector.