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In the present paper, we study convergence of the discrete theory back to the smooth theory when taking finer and finer triangulations, in the general Riemannian setting. We present suitable conditions on the weighted triangulations that ensure convergence of discrete harmonic maps to smooth harmonic maps, introducing the notion of (almost) asymptotically Laplacian weights, and we offer a systematic method to construct such weighted triangulations in the two-dimensional case. Our computer software <jats:monospace>Harmony<\/jats:monospace> successfully implements these methods to compute equivariant harmonic maps in the hyperbolic plane.<\/jats:p>","DOI":"10.4153\/s0008414x22000074","type":"journal-article","created":{"date-parts":[[2022,2,18]],"date-time":"2022-02-18T08:14:05Z","timestamp":1645172045000},"page":"531-580","update-policy":"http:\/\/dx.doi.org\/10.1017\/policypage","source":"Crossref","is-referenced-by-count":0,"title":["Computing harmonic maps between Riemannian manifolds"],"prefix":"10.4153","volume":"75","author":[{"given":"Jonah","family":"Gaster","sequence":"first","affiliation":[]},{"given":"Brice","family":"Loustau","sequence":"additional","affiliation":[]},{"given":"L\u00e9onard","family":"Monsaingeon","sequence":"additional","affiliation":[]}],"member":"2643","published-online":{"date-parts":[[2022,2,18]]},"reference":[{"key":"S0008414X22000074_r3","unstructured":"[3] Brunck, F. , Iterated medial triangle subdivision in surfaces of constant curvature. Preprint, 2021. arXiv:2107.04112"},{"key":"S0008414X22000074_r8","unstructured":"[8] Gaster, J. , Loustau, B. , and Monsaingeon, L. , Computing discrete equivariant harmonic maps. Preprint, 2018. arXiv:1810.11932"},{"key":"S0008414X22000074_r5","unstructured":"[5] de Saint-Gervais, H.-P. , Approximation d\u2019objets lisses par des objets PL. 2014\u20132019. http:\/\/analysis-situs.math.cnrs.fr\/Approximation-d-objets-lisses-par-des-objets-PL.html"},{"key":"S0008414X22000074_r9","doi-asserted-by":"publisher","DOI":"10.4153\/CJM-1967-062-6"},{"key":"S0008414X22000074_r11","doi-asserted-by":"publisher","DOI":"10.4310\/CAG.1997.v5.n2.a4"},{"key":"S0008414X22000074_r6","volume-title":"Harmonic maps between Riemannian polyhedra","volume":"142","author":"Eells","year":"2001"},{"key":"S0008414X22000074_r10","volume-title":"Harmonic mappings between Riemannian manifolds","volume":"4","author":"Jost","year":"1984"},{"key":"S0008414X22000074_r4","unstructured":"[4] Crane, K. , The $n$ -dimensional cotangent formula. 2019. https:\/\/www.cs.cmu.edu\/~kmcrane\/Projects\/Other\/nDCotanFormula.pdf"},{"key":"S0008414X22000074_r12","unstructured":"[12] Loustau, B. , Harmonic maps from K\u00e4hler manifolds. 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