Maximum norm a posteriori error estimate for a 3d singularly perturbed semilinear reaction-diffusion problem

Abstract

Abstract. A singularly perturbed semilinear reaction-diffusion problem in the unit cube, is discretized on arbitrary nonuniform tensor-product meshes. We establish a second-order maximum norm a posteriori error estimate that holds true uniformly in the small diffusion parameter. No mesh aspect ratio condition is imposed. This result is obtained by combining (i) sharp bounds on the Green's function of the continuous differential operator in the Sobolev W 1,1 and W 2,1 norms and (ii) a special representation of the residual in terms of an arbitrary current mesh and the current computed solution. Numerical results on a priori chosen meshes are presented that support our theoretical estimate.

Maximum Norm A Posteriori Error Estimation for Parabolic Problems Using Elliptic Reconstructions

“…But still of more interest are anisotropic meshes in the context of singularly perturbed differential equations (such as (1.1) with ε 3 1). For such equations, the maximum norm is sufficiently strong to capture sharp boundary and interior layers in their solutions, while locally anisotropic meshes (fine and anisotropic in layer regions and standard outside) have been shown to yield reliable numerical approximations in an efficient way (see, e.g., [3,8,12,18] and references therein). But such meshes are typically constructed a priori or by heuristic methods.…”

Maximum-norm a posteriori error estimates for singularly perturbed elliptic reaction-diffusion problems

“…Remark 2 Similar Green's function bounds for the case ε << 1 and C f ∼ 1, but on significantly simpler tensor-product domains are given in [7,24]. An inspection of the proofs in these papers reveals that in this case, all bounds of Theorem 1 are sharp with respect to ε, ρ and ρ.…”

“…The maximum norm, by contrast, is sufficiently strong to capture sharp layers in the exact solution, so it appears more suitable for such problems. A posteriori estimates in the maximum norm for equations of type (1.1) are given in [24,7]; the results are independent of the mesh aspect ratios, but apply only to tensor-product meshes. The situation with a priori error estimates in the maximum norm for such equations is much more satisfactory.…”

Maximum-Norm A Posteriori Error Estimates for Singularly Perturbed Reaction-Diffusion Problems on Anisotropic Meshes

“…A posteriori error estimates for a problem of type (1.1) on anisotropic meshes are also given in [3,9,13,14]. In [9,3] the error is also estimated in the maximum norm, but the considered meshes have a tensor-product structure, while [14,13] deal with general anisotropic meshes, but the error is estimated in a weaker energy norm.…”

“…In [9,3] the error is also estimated in the maximum norm, but the considered meshes have a tensor-product structure, while [14,13] deal with general anisotropic meshes, but the error is estimated in a weaker energy norm.…”