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I would like to use a numerical code (finite element, but question should also apply to finite difference methods) to solve a steady state heat problem in 2D. One input parameter is the maximum grid cell size, but I can’t find any reference to guide me in choosing this value. I searched some books on FE in my local library but they all start with a given value for the cell size. Is there some rule of thumb or better a design rule to select this value?

Cheers and thanks in advance!

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Unfortunately it's hard or impossible to guess a reasonable mesh parameter a priori. The typical approach is to start with some value and solve the problem. Next you pick a smaller value and compare the results. If the results differ a lot then you pick yet another smaller value and compare again.

Ultimately you end up with a small enough mesh parameter which seems to provide reasonable approximations to the original problem. It's always up to the user to decide what is 'reasonable'.

For simple problems you may be able to reduce the mesh parameter such that all further changes can be neglected up to the machine floating point precision. For three-dimensional problems you rarely can use such accuracy due to limited resources.

For some particular methods you can derive a strict upper bound to the error a posteriori using so called hypercircle estimate. Then you can decide your maximum error tolerance beforehands and 'refine' (reduce the mesh parameter) as long as this tolerance is obtained. This however is not yet used in many practical applications as far as I know.

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    Thanks a lot knl! My problem is picking some value to start with. I've tried 1/5, 1/10 or 1/20 of the shortest length but I've got mixed results. Is there some typical value in the literature? – 2017-02-07
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    There is no typical starting value. In some particular fields of engineering there are some rule of thumbs that are based on experience but since I don't know what you are solving it's impossible to say. What do you mean by mixed results? – 2017-02-12
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    I'm trying to solve a steady state temperature, that is to say I'm interested in solving Poisson's equation for the temperature of a solid. Mixed results in that with 1/10 for instance I require very few mesh refinements with some problems and with others I need to do a lot of refinements. – 2017-02-15
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    3D geometry? What type of elements are you using? Poisson equation is very well studied and you can also use automatic adaptive meshing if you have proper tools. – 2017-02-15