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Since the 1960's the finite element method emerged as a powerful tool for thenumerical simulation of countless physical phenomena or processes in appliedsciences. One of the reasons for this undeniable success is the greatversatility of the finite-element approach to deal with different types ofgeometries. This is particularly true of problems posed in curved domains ofarbitrary shape. In the case of function-value Dirichlet conditions prescribedon curvilinear boundaries method's isoparametric version for meshes consistingof curved triangles or tetrahedra has been mostly employed to recover theoptimal approximation properties known to hold for standard straight-edgedelements in the case of polygonal or polyhedral domains. However, besidesobvious algebraic and geometric inconveniences, the isoparametric technique ishelplessly limited in scope and simplicity, since its extension to degrees offreedom other than function values is not straightforward if not unknown. Thepurpose of this paper is to propose, study and test a simple alternative thatbypasses all the above drawbacks, without eroding qualitative approximationproperties. More specifically this technique can do without curved elements andis based only on polynomial algebra. REMARKS : First submission (dated Jan. 3,2017) updated on Jan. 11, 2017 with the addition of a footnote on author'sresearch grant. A third version with several improvements was posted on Nov. 2,2017. The fourth version incorporated some findings during the revision of arelated paper. In the fifth version, besides minor changes, theconvection-diffusion equation became the model problem; the text was reviewedin order to highlight the advantages of the new approach over classicaltechniques, more particularly by means of additional numerical examples.

Optimal simplex finite-element approximations of arbitrary order in curved domains circumventing the isoparametric technique