Thats a unique idea.You should consider the following idea to enhance your idea so that it would impact uniquely:  consider the approximation of a coupled system of two singularly perturbed reaction-diffusion equations, with the finite element method. The solution to such problems contains boundary layers which overlap and interact, and the numerical approximation must take this into account in order for the resulting scheme to converge uniformly with respect to the singular perturbation parameters.  present results on a high order hpfinite element scheme which includes elements of size O(Ɛp) and O(µp) near the boundary, where Ɛ, µ are the singular perturbation parameters and pis the degree of the approximating polynomials. Under the assumption of analytic input data, the method yields exponential rates of convergence as p→ ∞, independently of Ɛ and µ. Numerical computations supporting the theory are also presented.
['Christos Xenophontos']
I am writing a research paper on a idea The hp finite element method for singularly perturbed systems of reaction-diffusion equations
[{"from": "human", "value": "I am writing a research paper on a idea The hp finite element method for singularly perturbed systems of reaction-diffusion equations"}, {"from": "gpt", "value": "Thats a unique idea.You should consider the following idea to enhance your idea so that it would impact uniquely:  consider the approximation of a coupled system of two singularly perturbed reaction-diffusion equations, with the finite element method. The solution to such problems contains boundary layers which overlap and interact, and the numerical approximation must take this into account in order for the resulting scheme to converge uniformly with respect to the singular perturbation parameters.  present results on a high order hpfinite element scheme which includes elements of size O(\u0190p) and O(\u00b5p) near the boundary, where \u0190, \u00b5 are the singular perturbation parameters and pis the degree of the approximating polynomials. Under the assumption of analytic input data, the method yields exponential rates of convergence as p\u2192 \u221e, independently of \u0190 and \u00b5. Numerical computations supporting the theory are also presented."}]