Thats a unique idea.You should consider the following idea to enhance your idea so that it would impact uniquely:  investigate a ak space-time formulation of the heat equation and its use for the construction of a numerical scheme. The formulation is based on a known ak space-time formulation, with the difference that a pointwise component of the solution, which in other works is usually neglected, is now kept.  investigate the role of such a component by first using it to obtain a pointwise bound on the solution and then deploying it to construct a numerical scheme. The scheme obtained, besides being quasi-optimal in the L2 sense, is also pointwise superconvergent in the temporal nodes.  prove a priori error estimates and  present numerical experiments to empirically support our findings.
['Stig Larsson', 'Matteo Molteni']
I am writing a research paper on a idea Numerical solution of parabolic problems based on a weak space-time formulation
[{"from": "human", "value": "I am writing a research paper on a idea Numerical solution of parabolic problems based on a weak space-time formulation"}, {"from": "gpt", "value": "Thats a unique idea.You should consider the following idea to enhance your idea so that it would impact uniquely:  investigate a ak space-time formulation of the heat equation and its use for the construction of a numerical scheme. The formulation is based on a known ak space-time formulation, with the difference that a pointwise component of the solution, which in other works is usually neglected, is now kept.  investigate the role of such a component by first using it to obtain a pointwise bound on the solution and then deploying it to construct a numerical scheme. The scheme obtained, besides being quasi-optimal in the L2 sense, is also pointwise superconvergent in the temporal nodes.  prove a priori error estimates and  present numerical experiments to empirically support our findings."}]