The environmental situation on Mars does not allow a buried corpse to decompose. Lack of microbes on the martian surface means that decomposition of organic material (i.e. dead bodies) is not possible. To avoid an exponential growth of number of tombs along with a growing population, a different kind of funeral will be required.
How the Swedish do it
Dead bodies are freeze-dried with liquid nitrogen, then powdered by a vibration mechanism and finally buried on the cemetery, not too far beneth the surface. The roots of plants grow into the powder and bring the substances of the powder into the circles of eternity. This funeral technology has an esthetic aspect: If the grandchild asks "Where has my grandma gone?" you can tell him "She has turned into a beautiful rose...".
Can it be done on Mars?
The funeral can be very similar to the Swedish method. The dried powder may be buried in parts of greenhouses to act as a fetilizer, aiding growth of plants. For that reason the greenhouses might be devided to different parts, one for food production and one for biological oxygen production and funeral.
Another option is traditional cremation. Burning bodies (and other medical/biological waste for that matter) breaks down biomass, such as the organs, into vapour and ash. However large, dense parts of the body, such as hip bones, will be left over and requires further processing. To generate enough heat in the furness to cremate the dead - 760 to 1150 °C (1400 to 2100 °F) - large quantities of energy would be required. This may not be a viable option for early Mars colonies as energy will be at a premium.
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